README.md

spin liquids and unconventional ordered states by mVMC

What do these scripts do?

• Description of spin liquids and unconventional ordered states by fermionic wave functions such as the projected BCS states and the Slater-Jastrow-type wave functions
• Use mVMC with the generalized pairing wave functions that include the projected BCS states and the Slater-Jastrow-type wave functions

References

• mVMC
  • DOI:10.1143/JPSJ.77.114701
  • DOI:10.1016/j.cpc.2018.08.014
• "Dictionary" of spin liquid ansatze
  • DOI:10.1103/PhysRevB.65.165113
  • arXiv:0905.4854

Famous wave functions

(under construction)

• 1D Heisenberg model

  • Projected Fermi sea in 1D, ground state of the Haldane-Shastry model

    • Wave function
      • f_k = 1 (|k|<pi/2), f_k = 0 (|k|>pi/2)
      • Equivalent to \epsilon_k = - 2 \cos k, \Delta_k = 0, \mu_k = 0 (Note that constant shift is allowed in f_k since f_ii is arbitrary for the hard Gutzwiller projection)
    • See DOI:10.1103/PhysRevB.36.381 DOI:10.1103/PhysRevLett.59.1472 DOI:10.1103/PhysRevLett.60.635 DOI:10.1103/PhysRevLett.60.639

  • Ground state of the Majumdar-Ghosh model

    • Wave function
      • f_ij = 1 (|i-j|=1), f_ij = 0 otherwise
      • Equivalent to \epsilon_k = -2t \cos 2k, \Delta_k = 4\sqrt{2}t \cos k, \mu = 0, which results in f_k = \sqrt{2} \cos k
    • See DOI:10.1063/1.1664978 DOI:10.1063/1.1664979 arXiv:0905.4854

  • VBS order with translation invariant wave functions

    • Wave function
      • Choose \epsilon_k and \Delta_k so that \sqrt{(\epsilon_k)^2 + (\Delta_k)^2} > 0 for all k
      • Seems valid in 1D but not sure for higher dimensions
    • See DOI:10.1103/PhysRevLett.91.257005 DOI:10.1103/PhysRevB.93.125127

  • SU(N) case

    • See DOI:10.1103/PhysRevB.91.174427 DOI:10.5075/epfl-thesis-7309 https://github.com/jduf/VMC-SUN-Heisenberg

• 1D Hubbard model, t-J model

  • Tomonaga-Luttinger liquid

    • Wave function
      • Long-range part of the Jastrow factor affects the charge gap
    • See DOI:10.1103/PhysRevLett.94.026406 DOI:10.1103/PhysRevB.72.085121

  • Luther-Emery liquid

    • Wave function
      • Close the charge gap by choosing the Jastrow factor for metal
      • Open the spin gap by choosing \sqrt{(\epsilon_k)^2 + (\Delta_k)^2} > 0 for all k
    • See DOI:10.1103/PhysRevLett.94.026406 DOI:10.1103/PhysRevB.72.085121

• 2D Heisenberg model

  • Square lattice

    • Ground state ansatz for the pure square Heisenberg model

      • See DOI:10.1103/PhysRevB.37.3774 DOI:10.1103/PhysRevB.100.125131

    • Gapless Z2 spin liquid ansatz for the J1-J2 square Heisenberg model (J2:next nearest neighbor)

      • Wave function
        • Z2Azz13 spin liquid defined as \epsilon_k = 2t(\cos k_x + \cos k_y), \Delta_k = \Delta_{x^2-y^2} (\cos k_x - \cos k_y) + \Delta_{xy} \sin 2k_x \sin 2k_y
      • See DOI:10.1103/PhysRevB.66.235110 DOI:10.1103/PhysRevLett.87.097201 DOI:10.1103/PhysRevB.88.060402 DOI:10.1103/PhysRevB.100.125131 DOI:10.1103/PhysRevB.99.100405 nhscp2014, F.Becca

    • Projected Fermi sea

      • The state shows unexpected long-range magnetic order, obeys the are law in 2D
      • See DOI:10.1209/0295-5075/103/57002 DOI:10.1103/PhysRevLett.107.067202 DOI:10.1103/PhysRevB.93.125127

  • Triangular lattice

    • Ground state ansatz for the pure triangular Heisenberg model

      • (Short-range) 3-site order (120 Neel) is reproduced by a 2-site (not a 3-site!) unit cell structure
      • See DOI:10.1103/PhysRevLett.92.1570031 DOI:10.1103/PhysRevB.74.014408 DOI:10.1103/PhysRevB.80.012404 DOI:10.1103/PhysRevB.88.125135

    • Spin liquid ansatz for the J-J' triangular Heisenberg model (J':spatial anisotropy)

      • See arXiv:cond-mat/0210662 DOI:10.1103/PhysRevLett.92.1570031 DOI:10.1103/PhysRevB.74.014408 DOI:10.1103/PhysRevB.80.012404 DOI:10.1103/PhysRevB.93.085111

    • Gapless U(1) Dirac spin liquid ansatz for the J1-J2 triangular Heisenberg model (J2:next nearest neighbor)

      • See DOI:10.1103/PhysRevB.93.144411 DOI:10.1038/s41467-019-11727-3
      • See also DOI:10.7566/JPSJ.83.093707

    • Spin liquid with spinon Fermi surface, projected Fermi sea, ground state ansatz for the triangular Heisenberg model with the strong ring-exchange interaction

      • See DOI:10.1103/PhysRevB.72.045105

  • Honeycomb lattice

    • Spin liquid ansatz for the J1-J2 honeycomb Heisenberg model (J2:next nearest neighbor)
      • See DOI:10.1103/PhysRevB.96.104401

  • Kagome lattice

    • Gapped Z2 spin liquid ansatz

      • See DOI:10.1103/PhysRevB.84.020407 arXiv:1601.02165
      • See also (not VMC but) DOI:10.1126/science.1201080 DOI:10.1103/PhysRevLett.109.067201

    • Gapless U(1) Dirac spin liquid ansatz

      • See DOI:10.1103/PhysRevLett.98.117205 DOI:10.1103/PhysRevB.84.020407 DOI:10.1103/PhysRevB.91.020402 DOI:10.1038/s41467-019-11727-3

    • Other spin liquid ansatze

      • See DOI:10.1103/PhysRevB.99.100405

  • Square-Kagome lattice, Shuriken lattice

    • Spin liquid ansatze, dimer state ansatze
      • The ground state of the isotropic model seems to exhibit pinwheel VBC order
      • See arXiv:2110.08198

• 2D Hubbard model

  • Charge order with translation invariant wave functions
    • See DOI:10.1103/PhysRevB.93.125127

• 3D Heisenberg model

  • Pyrochlore lattice

    • Spin liquid ansatze, ansatze with inversion/rotation symmetry breaking
      • The ground state of the isotropic model seems to exhibit broken inversion symmetry
      • See DOI:10.1103/PhysRevLett.126.117204 (GS by DMRG) arXiv:2101.08787 (GS by VMC) DOI:10.1103/PhysRevB.78.180410 (VMC ansatze in Table.I) DOI:10.1103/PhysRevB.79.144432 (VMC ansatze in Table.I) arXiv:2110.08160 (GS by PFFRG, updated VMC ansatze in Table.I)