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SU(N), Spin S antiferromagnet

This Hamiltonian has been implemented for Schwab et al. (2023). It defines an SU(NN) antiferromagnetic spin-SS Heisenberg model, for representations given by a Young tableau with N/2N/ 2 rows and 2S2S columns (c.f. Figure 1. Specifically, the model is:

H^=J∑⟨i,j⟩,aS^i(a)S^j(a), \hat{H} = J\sum_{\langle i, j \rangle, a } \hat{S}^{(a)}_i\hat{S}^{(a)}_j,
(1)

where the sum extends over the pair of nearest-neighbor sites ⟨i,j⟩\langle i, j \rangle, and aa runs over N2−1N^2 - 1 generators of the said representation of the su(N)\mathfrak{su}(N) algebra of SU(NN).

Young tableau corresponding to the irreducible representation of \mathfrak{su}(N) considered here; N is even and S is semi-integer.

Figure 1:Young tableau corresponding to the irreducible representation of su(N)\mathfrak{su}(N) considered here; NN is even and SS is semi-integer.

The actually simulated Hamiltonian is

H^QMC=H^J+H^Casimir+H^U=−J4∑⟨i,j⟩,α,β{D^(i,α),(j,β),D^(i,α),(j,β)†}+JH2∑i∑α>β{D^(i,α),(i,β),D^(i,α),(i,β)†}+U∑i,α(n^i,α−N2)2,\begin{split} \hat{H}_{\text{QMC}} =&\hat{H}_J + \hat{H}_{\text{Casimir}} + \hat{H}_U \\ =&- \frac{J}{4} \sum_{\langle i,j \rangle, \alpha,\beta } \left\{ \hat{D}_{(i,\alpha),(j,\beta)}, \hat{D}^{\dagger}_{(i,\alpha),(j,\beta)} \right\} \\ &+ \frac{J_H}{2} \sum_{i} \sum_{\alpha>\beta} \left\{ \hat{D}_{(i,\alpha),(i,\beta)}, \hat{D}^{\dagger}_{(i,\alpha),(i,\beta)} \right\}\\ &+ U\sum_{i,\alpha}\left(\hat{n}_{i,\alpha}-\frac{N}{2}\right)^2,\\ \end{split}
(2)

where

D^(i,α),(j,β)≡∑σc^i,α,σ†c^j,β,σ†,{A^,B^}≡A^B^+B^A^,n^i,α≡∑σ=1Nc^i,α,σ†c^i,α,σ†.\hat{D}_{(i,\alpha),(j,\beta)} \equiv \sum_{\sigma} \hat{c}_{i,\alpha,\sigma}^{\dagger}\hat{c}_{j,\beta,\sigma}^{\phantom{\dagger}}, \qquad \left\{\hat{A}, \hat{B}\right\} \equiv \hat{A} \hat{B} + \hat{B} \hat{A}, \qquad \hat{n}_{i,\alpha} \equiv \sum_{\sigma=1}^{N} \hat{c}_{i,\alpha,\sigma}^{\dagger}\hat{c}_{i,\alpha,\sigma}^{\phantom{\dagger}}.
(3)

The terms with the parameters JHJ_{\text{H}} and UU are projecting unto the desired Hilbert space to achieve Eq. (1).

The following square lattice geometries are implemented:

  • periodic boundary conditions (Lattice_type = Square)

  • open boundary conditions in one dimension and periodic boundary conditions in the other (Lattice_type = N_leg_ladder)

  • open boundary conditions (Lattice_type = Open_square)

Extending to other lattices should be relatively easy. If the lattices are bipartite, the simulations will be sign-problem free.

The following model-specific parameters are available:

Character (len=64) :: Lattice_type = 'Square'  ! Implemented: Square, N_leg_ladder and Open_square
Integer :: L1 = 4                  ! Size of lattice in first dimension
Integer :: L2 = 4                  ! Size of lattice in second dimension
!Integer :: N_SUN = 2              ! SU(N) symmtry
Integer :: N_Spin = 1              ! 2*S
!logical :: Projector = .false.    ! Use projetive finite temperature method
!logical :: Symm = .false.         ! Use symmetric Trotter decomposition
real(dp) :: ham_J = 1.d0           ! Antiferromagnetic interaction strength
real(dp) :: ham_U = 1.d0           ! On-site Hubbard term for freezing out charge fluctuations
real(dp) :: ham_J2 = -1.d0         ! Projection parameter for going to fully symmetric representation
real(dp) :: dtau = 0.1d0           ! Imaginary time step
real(dp) :: beta = 5.d0            ! Reciprcal temperature
real(dp) :: theta = 0.d0           ! Zero-temperature projection parameter
real(dp) :: pinning_factor = 1.d0  ! Factor by which a single J-value will be multiplied to achieve phase pinning.
integer :: pinning_x = 0           ! x coordinate of pinned bond
integer :: pinning_y = 0           ! y coordinate of pinned bond

Usage example

A brief usage example using pyALF.

!./clone_alf.sh
Cloning into 'ALF'...
remote: Enumerating objects: 29841, done.
remote: Counting objects: 100% (296/296), done.
remote: Compressing objects: 100% (292/292), done.
remote: Total 29841 (delta 185), reused 0 (delta 0), pack-reused 29545 (from 1)
Receiving objects: 100% (29841/29841), 243.82 MiB | 92.40 MiB/s, done.
Resolving deltas: 100% (21204/21204), done.

Import necessary pyALF classes.

from py_alf import ALF_source, Simulation

Create ALF source object and inspect available parameters

alf_src = ALF_source(alf_dir='./ALF')
alf_src.get_ham_names()
['Kondo', 'Hubbard', 'Hubbard_Plain_Vanilla', 'tV', 'LRC', 'Z2_Matter', 'Spin_Peierls', 'AFM_N_S']
alf_src.get_default_params('AFM_N_S', include_generic=False)
OrderedDict([('VAR_AFM_N_S', {'Lattice_type': {'comment': 'Implemented: Square, N_leg_ladder and Open_square', 'defined_in_base': False, 'value': 'Square'}, 'L1': {'comment': 'Size of lattice in first dimension', 'defined_in_base': False, 'value': 4}, 'L2': {'comment': 'Size of lattice in second dimension', 'defined_in_base': False, 'value': 4}, 'N_SUN': {'comment': 'SU(N) symmtry', 'defined_in_base': True, 'value': 2}, 'N_Spin': {'comment': '2*S', 'defined_in_base': False, 'value': 1}, 'Projector': {'comment': 'Use projetive finite temperature method', 'defined_in_base': True, 'value': False}, 'Symm': {'comment': 'Use symmetric Trotter decomposition', 'defined_in_base': True, 'value': False}, 'ham_J': {'comment': 'Antiferromagnetic interaction strength', 'defined_in_base': False, 'value': 1.0}, 'ham_U': {'comment': 'On-site Hubbard term for freezing out charge fluctuations', 'defined_in_base': False, 'value': 1.0}, 'ham_J2': {'comment': 'Projection parameter for going to fully symmetric representation', 'defined_in_base': False, 'value': -1.0}, 'dtau': {'comment': 'Imaginary time step', 'defined_in_base': False, 'value': 0.1}, 'beta': {'comment': 'Reciprcal temperature', 'defined_in_base': False, 'value': 5.0}, 'theta': {'comment': 'Zero-temperature projection parameter', 'defined_in_base': False, 'value': 0.0}, 'pinning_factor': {'comment': 'Factor by which a single J-value will be multiplied to achieve phase pinning.', 'defined_in_base': False, 'value': 1.0}, 'pinning_x': {'comment': 'x coordinate of pinned bond', 'defined_in_base': False, 'value': 0}, 'pinning_y': {'comment': 'y coordinate of pinned bond', 'defined_in_base': False, 'value': 0}})])

Create and run Simulation

sim = Simulation(
    alf_src, "AFM_N_S",
    {
        'L1': 4,
        'L2': 4,
        'Projector': True,
        'Symm': True,
        'N_Spin': 2,
        'theta': 2.,
        'beta': 0.5,
     },
     machine='GNU',
     mpi=True,
    )
# On first compile, an ALF-local version of HDF5 will be compiled, 
# which takes several minutes and echos many warnings.
sim.compile()
Fetching long content....
Compiling ALF... 
Cleaning up Prog/
Cleaning up Libraries/
Cleaning up Analysis/
Compiling Libraries

ar: creating modules_90.a
ar: creating libqrref.a

Compiling Analysis
Compiling Program
Parsing Hamiltonian parameters
filenames: Hamiltonians/Hamiltonian_Kondo_smod.F90 Hamiltonians/Hamiltonian_Kondo_read_write_parameters.F90
filenames: Hamiltonians/Hamiltonian_Hubbard_smod.F90 Hamiltonians/Hamiltonian_Hubbard_read_write_parameters.F90
filenames: Hamiltonians/Hamiltonian_Hubbard_Plain_Vanilla_smod.F90 Hamiltonians/Hamiltonian_Hubbard_Plain_Vanilla_read_write_parameters.F90
filenames: Hamiltonians/Hamiltonian_tV_smod.F90 Hamiltonians/Hamiltonian_tV_read_write_parameters.F90
filenames: Hamiltonians/Hamiltonian_LRC_smod.F90 Hamiltonians/Hamiltonian_LRC_read_write_parameters.F90
filenames: Hamiltonians/Hamiltonian_Z2_Matter_smod.F90 Hamiltonians/Hamiltonian_Z2_Matter_read_write_parameters.F90
filenames: Hamiltonians/Hamiltonian_Spin_Peierls_smod.F90 Hamiltonians/Hamiltonian_Spin_Peierls_read_write_parameters.F90
filenames: Hamiltonians/Hamiltonian_AFM_N_S_smod.F90 Hamiltonians/Hamiltonian_AFM_N_S_read_write_parameters.F90
Compiling program modules
Link program
Done.

sim.run()
Prepare directory "/home/jonas/Programs/hamiltonians/Hamiltonians/AFM_N_S/ALF_data/AFM_N_S_L1=4_L2=4_Projector=True_Symm=True_N_Spin=2_theta=2.0_beta=0.5" for Monte Carlo run.
Create new directory.
Run /home/jonas/Programs/hamiltonians/Hamiltonians/AFM_N_S/ALF/Prog/ALF.out
 ALF Copyright (C) 2016 - 2022 The ALF project contributors
 This Program comes with ABSOLUTELY NO WARRANTY; for details see license.GPL
 This is free software, and you are welcome to redistribute it under certain conditions.
 gap:    4.9347005129783043E-004   4.9347005129783043E-004
 No initial configuration
References
  1. Schwab, J., Parisen Toldin, F., & Assaad, F. F. (2023). Phase diagram of the SU(𝑁) antiferromagnet of spin 𝑆 on a square lattice. Phys. Rev. B, 108(11), 115151. 10.1103/PhysRevB.108.115151
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