Hubbard model in canonical ensemble.

This Hamiltonian extends the standard implementation of the Hubbard model with energetically imposed constraints that for calculations in the canonical ensemble and fixed z-component of total spin.

In the above,

and

The default values of $N_{\text{Particle}}$ and $S^z$ are $S^z = 0$ and $N_{\text{Particle}} = N \cdot N_{\text{Orb}}$, corresponding to half-filling. Here $N$ corresponds to the number of unit cells and $N_{\text{Orb}}$ the number of orbitals per unit cell.

In the present implementation, the projection onto the $S^z =0$ subspace generates a mild sign problem. For the 4-site chain at $U/t=4$ and $\beta t = 2$ the QMC energy in the $S^z=0$ and $N=4$ sector gives $-1.8137 \pm 0.0035$ and the ED result is -1.8100613 so that the present version of the code seems to work.

Here are the required entries for the parameter file.

&VAR_ham_name
ham_name = "Hubbard_Can"
/

and

&VAR_Hubbard_Can            !! Variables for the specific model
Mz         = .F.            ! When true, sets the M_z-Hubbard model: Nf=2, demands that
                            ! N_sun is even, HS field couples to the z-component of
                            ! magnetization; otherwise, HS field couples to the density
Continuous = .F.            ! Uses (T: continuous; F: discrete) HS transformation
ham_T      = 1.d0           ! Hopping parameter
ham_chem   = 0.d0           ! Chemical potential
ham_U      = 4.d0           ! Hubbard interaction
ham_Lambda_c = 5.00         ! Projection Charge
ham_Lambda_s = 20.00        ! Projection Spin
ham_T2     = 1.d0           ! For bilayer systems
ham_U2     = 4.d0           ! For bilayer systems
ham_Tperp  = 1.d0           ! For bilayer systems
/