Electrically driven singlet-triplet transition in triangulene spin-1 chains
Department of Physics of Complex Systems, Eötvös Loránd University
MTA-BME Lendület Topology and Correlation Research Group, Budapest University of Technology and Economics
László Oroszlány
The Team
Gabriel Martinez
Oviedo
Amador Garcia Fuente
Oviedo
László Szunyogh
BME
Jaime Ferrer
Oviedo
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
Hubbard model and magnetism
\displaystyle {\hat {H}}=-t\sum _{ij,\sigma }\left({\hat {c}}_{i,\sigma }^{\dagger }{\hat {c}}_{j,\sigma }+ \mathrm{h.c.}\right)
+U\sum _{i}{\hat {n}}_{i\uparrow }{\hat {n}}_{i\downarrow }
Y. Claveau, B. Arnaud, S. Matteo; Eur. J. Phys. 35 035023 (2014)
J. Hubbard, Proc. R. Soc. A 277 237 (1964)
Competition of kinetic ( \(\rightarrow\) No magnetism )
and
interacting terms ( \(\rightarrow\) Yes magnetism )
Magnetism at the zig-zag edge
ribbon width
gap [eV]
Magda et al. Nature 514, 608 (2014)
U
Mapping total energy to simple spin cahin
O. V. Yazyev and M. I. Katsnelson,
Phys. Rev. Lett. 100,047209 (2008).
spin-wave stiffness:
2100 meVÅ\(^2\)
exchange constant:
105 meV
Socio-scientific interlude
vs.
SKKR
+ Wannier90
Zig-zag ribbon with SIESTA
\( D_{Yazyev}=2108 meVA^2 \)
\(\updownarrow\)
\( D_{edge}=2308 meVA^2 \)
\( D_{half}=3406 meVA^2 \)
Phys. Rev. B 99, 224412 (2019)
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_{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 )
DFT through
| Dimer | Infinite chain | Experiment | |
|---|---|---|---|
| 17.7 meV | 19.75 meV | 18 meV | |
| 0.03 | 0.05 | 0.09 |
The proof of the pudding
\( \beta \)
\( J \)
Singlet-Triplet qubit
M. D. Shulman, O. E. Dial, S. P. Harvey,H. Bluhm, V. Umansky, A. Yacoby; Science 336, 202 (2012)
Singlet-Triplet qubit in an S=1 chain
B. Jaworowski, N. Rogers, M. Grabowski, P. Hawrylak Sci. Rep. 7, 5529 (2017)
Singlet-Triplet transition in odd-length chains
Singlet-Triplet transition through mechanical distortions
But there is a dipole!
Singlet-Triplet transition through electric field
Thank You for your attention!
arXiv:2207.13683
| Phys. Rev. B 107, 035432 (2023) |
\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 )
Bolyai-lecture
By László Oroszlány
