QSH t-\(\lambda\) model

Model definition

The model is defined and discussed in this article [3] and is implemented as

\[ \hat{H} = -t\sum_{\langle\boldsymbol{i},\boldsymbol{j}\rangle}\sum_{\sigma=\uparrow,\downarrow}\sum_{s=1}^{N_{\textrm{SU}(N)}}\big(\hat{c}_{\boldsymbol{i},\sigma,s}^\dagger\hat{c}_{\boldsymbol{j},\sigma,s}+\textrm{h.c.}\big) -\frac{\lambda}{N_{\textrm{SU}(N)}}\sum_{\boldsymbol{r}_{\textrm{Hexagon}}}\Big(\sum_{\sigma,\sigma'}\sum_{s=1}^{N_{\textrm{SU}(N)}}\sum_{\langle\langle\boldsymbol{i},\boldsymbol{j}\rangle\rangle\in \boldsymbol{r}_{\textrm{Hexagon}}}i\nu_{\boldsymbol{i}\boldsymbol{j}}\hat{c}_{\boldsymbol{i},\sigma,s}^\dagger\boldsymbol{\sigma}_{\sigma,\sigma'}\hat{c}_{\boldsymbol{j},\sigma',s}+\textrm{h.c.}\Big)^2 \]

Physically, the implementation supports the honeycomb lattice. Note that the spin is encoded as layer such that under &VAR_lattice in the parameter file Lattice_type = "Bilayer_honeycomb" has to be chosen. The parameter file for this specific model reads:

&VAR_QSH              !! Variables for the specific model
ham_T      = 1.d0           ! Hopping parameter
ham_T2     = 1.d0
ham_chem   = 0.d0           ! Chemical potential
ham_lambda = 0.1d0
/

In the above Ham_T is the nearest neighbor hopping and Ham_lambda is the coupling strength. Finally Ham_chem is the chemical potential. To use this Hamiltonian you have to specify:

&VAR_ham_name
ham_name = "QSH"
/

in the parameters file.

Interaction

In order to reduce the Trotter error, the interaction term is implemented by using a Trotter decomposition corresponding to a Kekulé pattern by splitting up the lattice into three subgroups A,B,C. The specific implementation of the interaction is visually sketched in the smod.F90 file of the Hamiltonian.

Observables

The code has the standard observables as well as correlations of kinetic energy, quantum spin-Hall, s-wave pairing and correlations of the generators of SU(2). Note that the potential and total energies are defined as in the Hamiltonian. That is the file Ener_scalJ corresponds to \(\langle \hat{H} \rangle\) with \(\hat{H}\) defined as in the first equation.

Limitations

As it stands the trial wave function is not implemented such that only the finite temperature and not the projective code can be used. Therefore, the code only works for Projector = .F. under &VAR_Model_Generic.

[1]

Jonas Schwab, Francesco Parisen Toldin, and Fakher F. Assaad. Phase diagram of the SU(𝑁) antiferromagnet of spin 𝑆 on a square lattice. Phys. Rev. B, 108:115151, Sep 2023. arXiv:2304.07329, doi:10.1103/PhysRevB.108.115151.

[2]

Jonas Schwab, Lukas Janssen, Kai Sun, Zi Yang Meng, Igor F. Herbut, Matthias Vojta, and Fakher F. Assaad. Nematic Quantum Criticality in Dirac Systems. Phys. Rev. Lett., 128:157203, Apr 2022. arXiv:2110.02668, doi:10.1103/PhysRevLett.128.157203.

[3]

Gabriel Rein, Marcin Raczkowski, Zhenjiu Wang, Toshihiro Sato, and Fakher F. Assaad. Manipulating topology of quantum phase transitions by symmetry enhancement. 2024. URL: https://arxiv.org/abs/2410.05059, arXiv:2410.05059.