Spin 1 physics in molecules and quantum dots

Department of Physics of Complex Systems

Eötvös Loránd University

László Oroszlány

Haldane's conjecture and the AKLT model

\displaystyle \hat H = {\color{red}J} \left (\sum_p \vec{S}_p \cdot \vec{S}_{p+1} + {\color{red}\beta} (\vec{S}_p \cdot \vec{S}_{p+1})^2 \right )

BL-BQ model

\displaystyle \hat H = \sum_p \vec{S}_p \cdot \vec{S}_{p+1} + \frac{1}{3} (\vec{S}_p \cdot \vec{S}_{p+1})^2

Affleck, Kennedy, Lieb, Tasaki Phys. Rev. Lett. 59, 799 (1987)

\displaystyle H=\sum_p \vec{S}_p\cdot\vec{S}_{p+1}

\(S=\frac{1}{2}\rightarrow\) gapless

\(S=1\rightarrow\) gapped

F. D. M. Haldane Phys. Rev. Lett. 50, 1153 (1983)

How to realize S=1 and the Haldane phase

Renard, Regnault, Verdaguer, in

Magnetism: Molecules to Materials I: Models and Experiments

inelastic neutron-scattering experiments on bulk samples
interchain interaction
anisotropy

Can we realize with S=1 with carbon?

Su, Telychko, Song, Lu Angewandte Chemie 59 7658 (2020)

YES!

E. Lieb,

Phys. Rev. Lett. 62 1201 (1989)

Making a chain!

Mishra et al. Nature 598, 287 (2021).

fit to BL-BQ model:

J=18 \mathrm{meV}\, ,\beta=0.09

Kicking a black box to learn about its contents

\displaystyle \mathcal{H}=-\frac{1}{2}\sum_{i\ne j}J_{ij}\,\vec{e}_{i}\vec{e}_{j}

J_{ij}=?

\delta E(\vec{e}_{i})=E(\vec{e}_{i})-E_{0}=\left(1-\vec{e}_{i}\vec{e}_{0}\right)\sum\limits _{k\left(\neq i\right)}J_{ik}

\delta E(\vec{e}_{i},\vec{e}_{j})=E(\vec{e}_{i},\vec{e}_{j})-E_{0}=\left(1-\vec{e}_{i}\vec{e}_{j}\right)J_{ij} +\left(1-\vec{e}_{i}\vec{e}_{0}\right)\sum\limits _{k\left(\neq i,j\right)}J_{ik}+\left(1-\vec{e}_{j}\vec{e}_{0}\right)\sum\limits _{k\left(\neq i,j\right)}J_{jk}\, \\ =\delta E(\vec{e}_{i})+\delta E(\vec{e}_{j})-\left(\vec{e}_{i}-\vec{e}_{0}\right)\left(\vec{e}_{j}-\vec{e}_{0}\right)J_{ij}\,.

\displaystyle E_{0}=-\frac{1}{2}\sum_{i\ne j}J_{ij}

Assume a ferromagnetic arrangement:

E_{ij}^{\mathrm{int}} =\delta E(\vec{e}_{i},\vec{e}_{j})-\delta E(\vec{e}_{i})-\delta E(\vec{e}_{j}) =-J_{ij}\,\delta\vec{e}_{i}\delta\vec{e}_{j}

Ab initio spin model parameters

\displaystyle \delta E^{int}_{nm} = -\frac{1}{\pi} \int \mathrm{Im} \mathrm{Tr} [G\delta V_n G \delta V_m] \mathrm{d}E \rightarrow \mathbb{E}_{nm}\, \delta\mathbf{S}_n\cdot\delta\mathbf{S}_m

\mathbb{E}_{nm}=J_{nm}\,\left(\,1+2\,\beta_{nm}\left(\mathbf{S}_n\cdot\mathbf{S}_m\right)\,\right)

J_{nm} = \left(\mathbb{E}_{nm}^{FM}+ \mathbb{E}_{nm}^{AFM}\right)/2\\\,\\ \beta_{nm} = \dfrac{1}{2}\,\dfrac{\mathbb{E}_{nm}^{FM}- \mathbb{E}_{nm}^{AFM}}{\mathbb{E}_{nm}^{FM}+\mathbb{E}_{nm}^{AFM}}

\displaystyle \hat H = \sum_{nm} {\color{red}J_{nm}} \left (\vec{S}_n \cdot \vec{S}_{m} + {\color{red}\beta_{nm}} (\vec{S}_n \cdot \vec{S}_{m})^2 \right )