.. _03_excBHZ: In-plane spin excitons in QSHI ========================================================== In this finale example we discuss an interesting symmetry breaking case in the same QSHI model introduced in :doc:`01_bhz`. Specifically, we consider the case of excitonic condensation described by the order parameters :math:`\vec{E}=[E_0,E_1,E_2,E_3]` which, in terms of the Gamma matrices, read :math:`E_a = \langle \psi_i^\dagger \Gamma_{a1} \psi_i \rangle`. The analysis of the impurity susceptibilities in the previous section, suggest a possible instability towards the in-plane triplet exciton state :math:`E_1` or :math:`E_2`. Interestingly, the onset of this state breaks several symmetries, e.g. time-reversal and spin SU(2). As thus, this is a physical case to investigate the :f:var:`ed_mode` = **nonsu2** mode in |edipack2|. Source code ------------------------------ The considerations about the model Hamiltonian and the non-interacting solution remain identical to the previous case :doc:`01_bhz`. Here we discuss how to change the program to tackle the specific issue of in-plane exciton condensation in QSHI. The general structure of the code is unchanged but for one important part: the bath construction. We get: .. code-block:: fortran :linenos: !> Get local Hamiltonian summing over k (one can do better) allocate(Hloc(Nso,Nso)) Hloc = sum(Hk,dim=3)/Lk where(abs(dreal(Hloc))<1d-6)Hloc=zero !> Set H_{loc} in EDIpack2 call ed_set_hloc(Hloc) !> Get bath dimension and allocate user bath to this size ! ~removed~[Nb=ed_get_bath_dimension()]~ !> Setup the replica bath basis for the case E0EzEx(singlet,tripletZ,tripletX) allocate(lambdasym_vector(Nbath,4)) allocate(Hsym_basis(Nso,Nso,4)) Hsym_basis(:,:,1)=Gamma5 ;lambdasym_vector(:,1)= Mh Hsym_basis(:,:,2)=GammaE0 ;lambdasym_vector(:,2)= sb_field Hsym_basis(:,:,3)=GammaEz ;lambdasym_vector(:,3)= sb_field Hsym_basis(:,:,4)=GammaEx ;lambdasym_vector(:,4)=-sb_field !> Set the replica bath call ed_set_Hreplica(Hsym_basis,lambdasym_vector) !> Get bath dimension and allocate user bath to this size Nb=ed_get_bath_dimension(4) !(Hsym_basis) allocate(Bath(Nb)) ! !> Initialize the ED solver (bath is guessed or read from file) call ed_init_solver(bath) We first generate a basis of 4 matrices :math:`O_i=[\Gamma_{03},\Gamma_{01},\Gamma_{31},\Gamma_{11}]` and a set of parameters :math:`\vec{\lambda}^p=[ \lambda^p_1,\lambda^p_2,\lambda^p_3,\lambda^p_4]` which will be used to parametrize any bath Hamiltonian as :math:`h^p =\sum_i \lambda_i^p O_i`. The rest of the implementation is unaltered, except for a couple of printing flags. .. raw:: html
Results ------------------------------ We can now discuss some results obtained with this |edipack2| code concerning the exciton condensation in QSHI. A more thorough presentation can be found in `PhysRevB.107.115117`_. .. _PhysRevB.107.115117: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.115117 To make a connection with the non-ordered case, we consider here :math:`M=1` and fix the Hund's exchange to :math:`J/U=0.25`. Then we illustrate the effect of raising the interaction strength :math:`U`. In the top panels of the following figure we report the evolution (from left to right) of the spectral functions :math:`-\Im G_{a=1,2}(\omega)` for increasing :math:`U` in the three different phases of the system, respectively: QSHI (:math:`U=2`), Excitonic insulator (:math:`U=4`) and Mott insulator (:math:`U=6`). Quite interestingly the spectral functions in the first two cases reveal an almost identical gap size at low energy. Yet, a closer look at the topological Hamiltonian :math:`H_{top}` show the completely different the low-energy effective band structure. In panel D we show the expected behavior for a QSHI, featuring a band inversion around the topological gap and full spin symmetry. In panel E however, we observe a much larger energy separation and the breaking of spin degeneracy. .. image:: 03_excBHZ_fig1.svg :class: with-border :width: 800px In order to better assess the nature of the state interposing between the QSHI and the Mott insulator, we report in the next figure the behavior of the in-plane triplet excitonic order parameter :math:`\langle E_x \rangle`. As the critical strength at about :math:`U\simeq 4` is reached we observe a sharp discontinuous formation of a broken symmetry state with a finite order parameter. Further increasing the interaction strength the value of :math:`\langle E_x \rangle` decreases slowly towards zero signalling the continuous transition to a Mott state (which can be possibly unstable towards magnetic ordering, neglected in this calculation). For comparison we report in the same figure also the evolution of the orbital polarization :math:`\langle T_z\rangle` and of the correlation strength :math:`\theta` defined as the deviation from a constant, mean-field, behavior of the sele-energy function. The first quantity has a decreasing behavior with a small discontinuity at the QSHI to EI transition, while it continuously vanishes at the Mott transition. Interestingly, the correlation strength which is smoothly increasing in the QSHI, shows a sudden increase in the EI reaching large values to finally settle down to a linearly increasing behavior in the Mott state. .. image:: 03_excBHZ_fig2.svg :class: with-border :width: 800px .. raw:: html
The program to solve the main model can be found here: :download:`Exciton BHZ Code <03_excBHZ.f90>` A li of replica bath parameters used in the calculations reported in fig.1 are here: * Bath :math:`M=1`, :math:`J/U=0.25` and :math:`U=2` :download:`hamiltonian.restart ` * Bath :math:`M=1`, :math:`J/U=0.25` and :math:`U=4` :download:`hamiltonian.restart ` * Bath :math:`M=1`, :math:`J/U=0.25` and :math:`U=6` :download:`hamiltonian.restart ` Here is an example of input file used in the calculations above: :download:`InputFile `