en:tests

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en:tests [2011/10/10 13:38] 127.0.0.1 external edit |
en:tests [2019/06/04 21:40] (current) thomas |
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\end{equation*} | \end{equation*} | ||

The form of the groundstate energy will change when the external magnetic field \(h^z \leq h^z_c\) is larger than the antiferromagnetic interaction and it is energetically cheaper for the system to align in direction of the magnetic field. The special intermediate case is just reasoned in the open boundary conditions as the spins at the end of the chain will align in the direction of the magnetic field before the inner spins (for a chain with an even number of sites). You can easily identify the three sectors in the following figure: | The form of the groundstate energy will change when the external magnetic field \(h^z \leq h^z_c\) is larger than the antiferromagnetic interaction and it is energetically cheaper for the system to align in direction of the magnetic field. The special intermediate case is just reasoned in the open boundary conditions as the spins at the end of the chain will align in the direction of the magnetic field before the inner spins (for a chain with an even number of sites). You can easily identify the three sectors in the following figure: | ||

- | [{{ :de:energie_gerade_ungerade.png?900 |grounstate energy depending on the external magnetic field for a chain with an even/odd number of sites}}] | + | [{{ :wiki:energie_gerade_ungerade.png?900 |grounstate energy depending on the external magnetic field for a chain with an even/odd number of sites}}] |

- | [{{ :de:sz_sx.png?900 |Expectation value of \(S^z\) depending on the external magnetic field for a chain with an even/odd number of sites}}] | + | [{{ :wiki:sz_sx.png?900 |Expectation value of \(S^z\) depending on the external magnetic field for a chain with an even/odd number of sites}}] |

==== Ferromagnetism ==== | ==== Ferromagnetism ==== | ||

The sign of the external magnetic field \(h^z\) gives the direction of the spins. | The sign of the external magnetic field \(h^z\) gives the direction of the spins. | ||

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with \(\Lambda_k^2=1+\lambda^2+ \lambda \cos(k)\) proportional to an elliptical integral of first kind. In the figures we show the results of the MPS-simulation with finite-size-scaling and the analytical solution. | with \(\Lambda_k^2=1+\lambda^2+ \lambda \cos(k)\) proportional to an elliptical integral of first kind. In the figures we show the results of the MPS-simulation with finite-size-scaling and the analytical solution. | ||

- | [{{ :de:energie.png?900 |Groundstate energy for the infinite system (after finite-size-scaling) depending on the magnetic field }}] | + | [{{ :wiki:energie.png?900 |Groundstate energy for the infinite system (after finite-size-scaling) depending on the magnetic field }}] |

- | [{{ :de:erwartungswert_z.png?900 |Expectation value \(S^z\) for the infinite system (after finite-size-scaling) depending on the magnetic field}}] | + | [{{ :wiki:erwartungswert_z.png?900 |Expectation value \(S^z\) for the infinite system (after finite-size-scaling) depending on the magnetic field}}] |

===== Heisenberg-Model ===== | ===== Heisenberg-Model ===== | ||

Within the Heisenberg-model the spins are not limited to specific direction anymore. They can point in any of the three space directions. The Hamiltonian of the Heisenberg-chain in one dimension is therefore given by: | Within the Heisenberg-model the spins are not limited to specific direction anymore. They can point in any of the three space directions. The Hamiltonian of the Heisenberg-chain in one dimension is therefore given by: |

en/tests.txt · Last modified: 2019/06/04 21:40 by thomas