Adel Abbout
\text{Adel Abbout}
King Fahd University of Petroleum and Minerals, Saudi Arabia
\text{King Fahd University of Petroleum and Minerals, Saudi Arabia}
\text{}
San Sebastian, 2023
\text{San Sebastian, 2023}
J(q)=−J1(2cos(qxa)+4cos(2qxa)cos(23qya))−J2(2cos(3qya)+4cos(23qxa)cos(23qya))
\displaystyle
\begin{aligned}
J(\mathbf{q})= & -J_1\left(2 \cos \left(q_x a\right)+4 \cos \left(\frac{q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right) \\
& -J_2\left(2 \cos \left(\sqrt{3} q_y a\right)+4 \cos \left(\frac{3 q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right)
\end{aligned}
H=−J1⟨i,j⟩∑Si⋅Sj−J2⟨⟨i,j⟩⟩∑SiSj
\displaystyle \mathcal{H}=-J_1 \sum_{\langle i, j\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j-J_2 \sum_{\langle\langle i, j\rangle\rangle} \boldsymbol{S}_i\boldsymbol{S}_j
Heisenberg Hamiltonian on Triangular lattice
\text { Heisenberg Hamiltonian on Triangular lattice }
Ground state depends on α=J1J2
\text { Ground state depends on } \displaystyle {\textcolor{red}{\alpha=\frac{J_2}{J_1}}}
Fourier transform
\text{Fourier transform}
Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R)
\text { Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R) }
Ground state depends on α=J1J2
\text { Ground state depends on } \displaystyle {\textcolor{red}{\alpha=\frac{J_2}{J_1}}}
α<81
\displaystyle \alpha<\frac{1}{8}
Minimum at A
\text { Minimum at } {\textcolor{red}A}
120∘ structure
120^\circ \text { structure }
Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R)
\text { Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R) }
J(q)=−J1(2cos(qxa)+4cos(2qxa)cos(23qya))−J2(2cos(3qya)+4cos(23qxa)cos(23qya))
\displaystyle
\begin{aligned}
J(\mathbf{q})= & -J_1\left(2 \cos \left(q_x a\right)+4 \cos \left(\frac{q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right) \\
& -J_2\left(2 \cos \left(\sqrt{3} q_y a\right)+4 \cos \left(\frac{3 q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right)
\end{aligned}
H=−J1⟨i,j⟩∑Si⋅Sj−J2⟨⟨i,j⟩⟩∑SiSj
\displaystyle \mathcal{H}=-J_1 \sum_{\langle i, j\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j-J_2 \sum_{\langle\langle i, j\rangle\rangle} \boldsymbol{S}_i\boldsymbol{S}_j
Heisenberg Hamiltonian on Triangular lattice
\text { Heisenberg Hamiltonian on Triangular lattice }
Ground state depends on α=J1J2
\text { Ground state depends on } \displaystyle {\textcolor{red}{\alpha=\frac{J_2}{J_1}}}
α<81
\displaystyle \alpha<\frac{1}{8}
Minimum at A
\text { Minimum at } {\textcolor{red}A}
120∘ structure
120^\circ \text { structure }
Fourier transform
\text{Fourier transform}
Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R)
\text { Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R) }
81<α<1
\displaystyle \frac{1}{8}<\alpha<1
Minimum at B. The vector Q is half the reciprocal unit vector.
\text { Minimum at } {\textcolor{blue}B} \text {. The vector } Q \text { is half the reciprocal unit vector. }
Ground state depends on α=J1J2
\text { Ground state depends on } \displaystyle {\textcolor{red}{\alpha=\frac{J_2}{J_1}}}
α<81
\displaystyle \alpha<\frac{1}{8}
Minimum at A
\text { Minimum at } {\textcolor{red}A}
120∘ structure
120^\circ \text { structure }
Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R)
\text { Th. Jolicoeur et al. Phys. Rev. B 42, 4800(R) }
81<α<1
\displaystyle \frac{1}{8}<\alpha<1
Minimum at B. The vector Q is half the reciprocal unit vector.
\text { Minimum at } {\textcolor{blue}B} \text {. The vector } Q \text { is half the reciprocal unit vector. }
Row-wise AF
\text { Row-wise AF}
Ground state depends on α=J1J2
\text { Ground state depends on } \displaystyle {\textcolor{red}{\alpha=\frac{J_2}{J_1}}}
α<81
\displaystyle \alpha<\frac{1}{8}
Minimum at A
\text { Minimum at } {\textcolor{red}A}
120∘ structure
120^\circ \text { structure }
81<α<1
\displaystyle \frac{1}{8}<\alpha<1
Minimum at B. The vector Q is half the reciprocal unit vector.
\text { Minimum at } {\textcolor{blue}B} \text {. The vector } Q \text { is half the reciprocal unit vector. }
Row-wise AF
\text { Row-wise AF}
α>1
\displaystyle \alpha>1
Minimum at C
\text { Minimum at } {\textcolor{green}C}
Incommensurate spiral
\text { Incommensurate spiral}
J(q)=−J1(2cos(qxa)+4cos(2qxa)cos(23qya))−J2(2cos(3qya)+4cos(23qxa)cos(23qya))
\displaystyle
\begin{aligned}
J(\mathbf{q})= & -J_1\left(2 \cos \left(q_x a\right)+4 \cos \left(\frac{q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right) \\
& -J_2\left(2 \cos \left(\sqrt{3} q_y a\right)+4 \cos \left(\frac{3 q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right)
\end{aligned}
H=−J1⟨i,j⟩∑Si⋅Sj−J2⟨⟨i,j⟩⟩∑Si⋅Sj
\displaystyle \mathcal{H}=-J_1 \sum_{\langle i, j\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j-J_2 \sum_{\langle\langle i, j\rangle\rangle} \boldsymbol{S}_i\cdot\boldsymbol{S}_j
Heisenberg Hamiltonian on Triangular lattice
\text { Heisenberg Hamiltonian on Triangular lattice }
Ground state depends on α=J1J2
\text { Ground state depends on } \displaystyle {\textcolor{red}{\alpha=\frac{J_2}{J_1}}}
Fourier transform
\text{Fourier transform}
J(q)=−J1(2cos(qxa)+4cos(2qxa)cos(23qya))−J2(2cos(3qya)+4cos(23qxa)cos(23qya))
\displaystyle
\begin{aligned}
J(\mathbf{q})= & -J_1\left(2 \cos \left(q_x a\right)+4 \cos \left(\frac{q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right) \\
& -J_2\left(2 \cos \left(\sqrt{3} q_y a\right)+4 \cos \left(\frac{3 q_x a}{2}\right) \cos \left(\frac{\sqrt{3} q_y a}{2}\right)\right)
\end{aligned}
H=−J1⟨i,j⟩∑Si⋅Sj−J2⟨⟨i,j⟩⟩∑Si⋅Sj
\displaystyle \mathcal{H}=-J_1 \sum_{\langle i, j\rangle} \boldsymbol{S}_i \cdot \boldsymbol{S}_j-J_2 \sum_{\langle\langle i, j\rangle\rangle} \boldsymbol{S}_i\cdot\boldsymbol{S}_j
Heisenberg Hamiltonian on Triangular lattice
\text { Heisenberg Hamiltonian on Triangular lattice }
Ground state depends on α=J1J2
\text { Ground state depends on } \displaystyle {\textcolor{red}{\alpha=\frac{J_2}{J_1}}}
Fourier transform
\text{Fourier transform}
4-spin interaction
\text{4-spin interaction}
Or using dynamical matrix
\text{Or using dynamical matrix}
Papa B. Ndiaye, Adel Abbout, Durga Goli, and Aureˊlien Manchon Phys. Rev. B 100, 144440
\text{Papa B. Ndiaye, Adel Abbout, Durga Goli, and Aurélien Manchon
Phys. Rev. B 100, 144440}
Postdoc positions are available
\text{\textcolor{red}{Postdoc positions }are available}
Topological Systems and Quantum Computing
\text{Topological Systems and Quantum Computing}
competitive salary
\text{competitive salary}
housing
\text{housing}
health care
\text{health care}
annual flight tickets
\text{annual flight tickets}
A d e l A b b o u t \text{Adel Abbout} K i n g F a h d U n i v e r s i t y o f P e t r o l e u m a n d M i n e r a l s , S a u d i A r a b i a \text{King Fahd University of Petroleum and Minerals, Saudi Arabia} \text{} S a n S e b a s t i a n , 2 0 2 3 \text{San Sebastian, 2023}
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