Susceptibility of the Hubbard-Zeeman model
In this tutorial we will compute the dynamic, momentum-resolved susceptibility \(\chi_{abcd}(\omega,\mathbf{q})\) of the Hubbard-Zeeman model, along the lines of Communications Physics 6, 289 (2023). Unlike previous tutorials, in this tutorial we will compute and look at the full tensor structure of the susceptibility. Both the BSE and DBSE will be used.
The model has a hopping parameter :math: t, a Hubbard interaction :math: U, and a Zeeman magnetic field :math: B. The latter appears in the Hamiltonian as \(-B (\hat{n}_\uparrow-\hat{n}_\downarrow)\) The Zeeman field lifts the degeneracy between spin up and spin down and leads to a richer spin structure of the susceptibility including non-trivial off-diagonal elements. TRIQS/cthyb is used as an impurity solver for the DMFT self-consistency loop, and then the vertices are calculated using W2Dynamics. The overall structure of the scripts is similar to previous tutorials: common.py is a module with general functionality, calc_sc_dmft.py runs the DMFT self-consistency loop and then calc_g2.py, calc_tri.py, calc_chi.py measure the three required correlation functions. Finally, calc_bse.py evaluates the Bethe-Salpeter equation with different fermionic cut-offs nwf.
Vertices
plot_vertex.py
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Convergence of Susceptibility
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Larmor precession
In the presence of a Zeeman field \(B\), the total spin of the system will undergo Larmor precession. This is visible in the in-plane (x,y) components of the spin susceptibility at \(\mathbf{q}=\Gamma\). The frequency of the oscillation is fixed by \(B\) and is thus known analytically. This is a good check on the numerical results.
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