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rad1um
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Consider the classical Heisenberg model without an external field which is defined by the Hamiltonian:
[tex] \mathcal{H} = -\sum_{ij} J_{ij} \vec{s_i}\vec{s_j} [/tex]
where [tex]J_{ij} > 0[/tex] describes the coupling between the spins [tex] \vec{s}_i \in \mathbb{R}^3 [/tex] on some lattice. (Is there a way to use tex inline?)
In my undergad courses we usually analised this model when coupled to a heat bath e.g. by usiang a Metropolis algorithm. We can however write down the equation of motion using Hamilton formalism leading to
[tex] \dot{\vec{s}_i} = \{ \vec{s}_i, \mathcal{H}\} = \frac{\partial \mathcal{H}}{\partial \vec{s}_i} \times \vec{s}_i = -\left( \sum_j J_{ij} \vec{s}_j \right) \times \vec{s}_i.[/tex]
My understanding is that this ODE describes the behavior of spins w/o a heat bath i.e. at T=0. This would mean that the fixed point of the system is the case where all spins align. In fact we can easily see that this is a fixed point of the system, since
[tex] \vec{s}_{||} \times \vec{s}_{||} = \vec{0} \text{ and thus } \dot{\vec{s_i}} = \vec{0}.[/tex]
However, when numerically solving this model (e.g. for a lattice with 3 sites where all sites are neighbors) I found that the fixed point is highly unstable and that the system has chaotic features. At first I thought my implementation was wrong, but I also read a few papers where the authors found positive Lyapunov exponents consequencing un-stability (see here for example: https://arxiv.org/pdf/1209.1468.pdf).
What makes this even weirder for me is that we can transform the EOM of the Heisenberg model to the the Kuramoto model, like so
[tex] \left | \dot{\vec{s}_i} \right| = \sum_j J_{ij} \left| \vec{s}_j \times \vec{s}_i \right| = \sum_j J_{ij} \mathrm{sin}(\phi_j - \phi_i) = \dot{\phi_i}[/tex]
where [tex] \phi_j - \phi_i [/tex] is the difference of angles between two spins. And the Kuramoto model has a stable fixed point for all [tex] \Delta \phi_i = 0, [/tex] which would translate to full alignmed of all spins.
Somewhere I am doing a mistake. It would be awesome if someone could enlighten me.
Have a nice day!
[tex] \mathcal{H} = -\sum_{ij} J_{ij} \vec{s_i}\vec{s_j} [/tex]
where [tex]J_{ij} > 0[/tex] describes the coupling between the spins [tex] \vec{s}_i \in \mathbb{R}^3 [/tex] on some lattice. (Is there a way to use tex inline?)
In my undergad courses we usually analised this model when coupled to a heat bath e.g. by usiang a Metropolis algorithm. We can however write down the equation of motion using Hamilton formalism leading to
[tex] \dot{\vec{s}_i} = \{ \vec{s}_i, \mathcal{H}\} = \frac{\partial \mathcal{H}}{\partial \vec{s}_i} \times \vec{s}_i = -\left( \sum_j J_{ij} \vec{s}_j \right) \times \vec{s}_i.[/tex]
My understanding is that this ODE describes the behavior of spins w/o a heat bath i.e. at T=0. This would mean that the fixed point of the system is the case where all spins align. In fact we can easily see that this is a fixed point of the system, since
[tex] \vec{s}_{||} \times \vec{s}_{||} = \vec{0} \text{ and thus } \dot{\vec{s_i}} = \vec{0}.[/tex]
However, when numerically solving this model (e.g. for a lattice with 3 sites where all sites are neighbors) I found that the fixed point is highly unstable and that the system has chaotic features. At first I thought my implementation was wrong, but I also read a few papers where the authors found positive Lyapunov exponents consequencing un-stability (see here for example: https://arxiv.org/pdf/1209.1468.pdf).
What makes this even weirder for me is that we can transform the EOM of the Heisenberg model to the the Kuramoto model, like so
[tex] \left | \dot{\vec{s}_i} \right| = \sum_j J_{ij} \left| \vec{s}_j \times \vec{s}_i \right| = \sum_j J_{ij} \mathrm{sin}(\phi_j - \phi_i) = \dot{\phi_i}[/tex]
where [tex] \phi_j - \phi_i [/tex] is the difference of angles between two spins. And the Kuramoto model has a stable fixed point for all [tex] \Delta \phi_i = 0, [/tex] which would translate to full alignmed of all spins.
Somewhere I am doing a mistake. It would be awesome if someone could enlighten me.
Have a nice day!
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