Néel order in the $U(1)$-symmetric XXZ model
Here, we want to look at a special case of the Heisenberg model, where the $x$ and $y$ couplings are equal, called the XXZ model
\[H_0 = J \big(\sum_{\langle i, j \rangle} S_i^x S_j^x + S_i^y S_j^y + \Delta S_i^z S_j^z \big) .\]
For appropriate $\Delta$, the model enters an antiferromagnetic phase (Néel order) which we will force by adding staggered magnetic charges to $H_0$. Furthermore, since the XXZ Hamiltonian obeys a $U(1)$ symmetry, we will make use of that and work with $U(1)$-symmetric PEPS and CTMRG environments. For simplicity, we will consider spin-$1/2$ operators.
But first, let's make this example deterministic and import the required packages:
using Random
using TensorKit, PEPSKit
using MPSKit: add_physical_charge
Random.seed!(2928528935);Constructing the model
Let us define the $U(1)$-symmetric XXZ Hamiltonian on a $2 \times 2$ unit cell with the parameters:
J = 1.0
Delta = 1.0
spin = 1//2
symmetry = U1Irrep
lattice = InfiniteSquare(2, 2)
H₀ = heisenberg_XXZ(ComplexF64, symmetry, lattice; J, Delta, spin);This ensures that our PEPS ansatz can support the bipartite Néel order. As discussed above, we encode the Néel order directly in the ansatz by adding staggered auxiliary physical charges:
S_aux = [
U1Irrep(-1//2) U1Irrep(1//2)
U1Irrep(1//2) U1Irrep(-1//2)
]
H = add_physical_charge(H₀, S_aux);Specifying the symmetric virtual spaces
Before we create an initial PEPS and CTM environment, we need to think about which symmetric spaces we need to construct. Since we want to exploit the global $U(1)$ symmetry of the model, we will use TensorKit's U1Spaces where we specify dimensions for each symmetry sector. From the virtual spaces, we will need to construct a unit cell (a matrix) of spaces which will be supplied to the PEPS constructor. The same is true for the physical spaces, which can be extracted directly from the Hamiltonian LocalOperator:
V_peps = U1Space(0 => 2, 1 => 1, -1 => 1)
V_env = U1Space(0 => 6, 1 => 4, -1 => 4, 2 => 2, -2 => 2)
virtual_spaces = fill(V_peps, size(lattice)...)
physical_spaces = physicalspace(H)2×2 Matrix{TensorKit.GradedSpace{TensorKitSectors.U1Irrep, TensorKit.SortedVectorDict{TensorKitSectors.U1Irrep, Int64}}}:
Rep[TensorKitSectors.U₁](0=>1, 1=>1) Rep[TensorKitSectors.U₁](0=>1, -1=>1)
Rep[TensorKitSectors.U₁](0=>1, -1=>1) Rep[TensorKitSectors.U₁](0=>1, 1=>1)Ground state search
From this point onwards it's business as usual: Create an initial PEPS and environment (using the symmetric spaces), specify the algorithmic parameters and optimize:
boundary_alg = (; tol=1e-8, alg=:simultaneous, trscheme=(; alg=:fixedspace))
gradient_alg = (; tol=1e-6, alg=:eigsolver, maxiter=10, iterscheme=:diffgauge)
optimizer_alg = (; tol=1e-4, alg=:lbfgs, maxiter=85, ls_maxiter=3, ls_maxfg=3)
peps₀ = InfinitePEPS(randn, ComplexF64, physical_spaces, virtual_spaces)
env₀, = leading_boundary(CTMRGEnv(peps₀, V_env), peps₀; boundary_alg...);[ Info: CTMRG init: obj = -2.356413456811e+03 +3.307968169629e+02im err = 1.0000e+00
[ Info: CTMRG conv 30: obj = +6.245129734283e+03 -4.010098564322e-08im err = 5.3638617277e-09 time = 0.92 sec
Finally, we can optimize the PEPS with respect to the XXZ Hamiltonian. Note that the optimization might take a while since precompilation of symmetric AD code takes longer and because symmetric tensors do create a bit of overhead (which does pay off at larger bond and environment dimensions):
peps, env, E, info = fixedpoint(
H, peps₀, env₀; boundary_alg, gradient_alg, optimizer_alg, verbosity=3
)
@show E;[ Info: LBFGS: initializing with f = -0.034628402377, ‖∇f‖ = 3.0490e-01
┌ Warning: Linesearch not converged after 1 iterations and 4 function evaluations:
│ α = 2.50e+01, dϕ = -6.00e-03, ϕ - ϕ₀ = -1.13e-01
└ @ OptimKit ~/.julia/packages/OptimKit/G6i79/src/linesearches.jl:148
[ Info: LBFGS: iter 1, time 200.41 s: f = -0.147314472246, ‖∇f‖ = 9.2219e-01, α = 2.50e+01, m = 0, nfg = 4
┌ Warning: Linesearch not converged after 1 iterations and 4 function evaluations:
│ α = 2.50e+01, dϕ = -1.99e-03, ϕ - ϕ₀ = -3.80e-01
└ @ OptimKit ~/.julia/packages/OptimKit/G6i79/src/linesearches.jl:148
[ Info: LBFGS: iter 2, time 269.67 s: f = -0.527654857567, ‖∇f‖ = 7.5002e-01, α = 2.50e+01, m = 0, nfg = 4
[ Info: LBFGS: iter 3, time 286.74 s: f = -0.552751928887, ‖∇f‖ = 3.6310e-01, α = 1.00e+00, m = 1, nfg = 1
[ Info: LBFGS: iter 4, time 337.91 s: f = -0.617206129929, ‖∇f‖ = 3.0891e-01, α = 3.20e+00, m = 2, nfg = 3
[ Info: LBFGS: iter 5, time 354.97 s: f = -0.632819487274, ‖∇f‖ = 4.0073e-01, α = 1.00e+00, m = 3, nfg = 1
[ Info: LBFGS: iter 6, time 370.30 s: f = -0.653138513637, ‖∇f‖ = 1.0717e-01, α = 1.00e+00, m = 4, nfg = 1
[ Info: LBFGS: iter 7, time 381.55 s: f = -0.655569926077, ‖∇f‖ = 4.7861e-02, α = 1.00e+00, m = 5, nfg = 1
[ Info: LBFGS: iter 8, time 392.14 s: f = -0.656621018321, ‖∇f‖ = 4.3322e-02, α = 1.00e+00, m = 6, nfg = 1
[ Info: LBFGS: iter 9, time 402.41 s: f = -0.657996751278, ‖∇f‖ = 4.6277e-02, α = 1.00e+00, m = 7, nfg = 1
[ Info: LBFGS: iter 10, time 412.44 s: f = -0.659837382098, ‖∇f‖ = 4.8930e-02, α = 1.00e+00, m = 8, nfg = 1
[ Info: LBFGS: iter 11, time 421.91 s: f = -0.661058228215, ‖∇f‖ = 4.1864e-02, α = 1.00e+00, m = 9, nfg = 1
[ Info: LBFGS: iter 12, time 431.77 s: f = -0.661633890101, ‖∇f‖ = 1.7067e-02, α = 1.00e+00, m = 10, nfg = 1
[ Info: LBFGS: iter 13, time 440.37 s: f = -0.661839831812, ‖∇f‖ = 1.4709e-02, α = 1.00e+00, m = 11, nfg = 1
[ Info: LBFGS: iter 14, time 449.27 s: f = -0.662134636143, ‖∇f‖ = 1.7065e-02, α = 1.00e+00, m = 12, nfg = 1
[ Info: LBFGS: iter 15, time 458.85 s: f = -0.662532716659, ‖∇f‖ = 1.7719e-02, α = 1.00e+00, m = 13, nfg = 1
[ Info: LBFGS: iter 16, time 467.98 s: f = -0.662996909612, ‖∇f‖ = 2.0543e-02, α = 1.00e+00, m = 14, nfg = 1
[ Info: LBFGS: iter 17, time 477.51 s: f = -0.663439633406, ‖∇f‖ = 2.6049e-02, α = 1.00e+00, m = 15, nfg = 1
[ Info: LBFGS: iter 18, time 486.45 s: f = -0.663855589525, ‖∇f‖ = 3.2567e-02, α = 1.00e+00, m = 16, nfg = 1
[ Info: LBFGS: iter 19, time 495.46 s: f = -0.664450880841, ‖∇f‖ = 2.3691e-02, α = 1.00e+00, m = 17, nfg = 1
[ Info: LBFGS: iter 20, time 505.64 s: f = -0.664917088518, ‖∇f‖ = 1.8281e-02, α = 1.00e+00, m = 18, nfg = 1
[ Info: LBFGS: iter 21, time 514.58 s: f = -0.665079077676, ‖∇f‖ = 1.2380e-02, α = 1.00e+00, m = 19, nfg = 1
[ Info: LBFGS: iter 22, time 523.59 s: f = -0.665287463007, ‖∇f‖ = 1.7237e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 23, time 533.23 s: f = -0.665717355832, ‖∇f‖ = 2.9356e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 24, time 543.00 s: f = -0.666166129443, ‖∇f‖ = 3.2946e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 25, time 562.76 s: f = -0.666407108756, ‖∇f‖ = 2.6211e-02, α = 4.29e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 26, time 573.09 s: f = -0.666612715340, ‖∇f‖ = 1.1288e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 27, time 582.76 s: f = -0.666709056953, ‖∇f‖ = 1.2143e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 28, time 592.05 s: f = -0.666869683924, ‖∇f‖ = 1.8077e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 29, time 602.27 s: f = -0.667130274911, ‖∇f‖ = 2.1430e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 30, time 622.23 s: f = -0.667250996806, ‖∇f‖ = 2.4838e-02, α = 4.17e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 31, time 631.86 s: f = -0.667422880816, ‖∇f‖ = 1.0123e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 32, time 641.47 s: f = -0.667499816879, ‖∇f‖ = 7.8017e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 33, time 651.74 s: f = -0.667576990274, ‖∇f‖ = 1.2346e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 34, time 661.42 s: f = -0.667710803542, ‖∇f‖ = 1.5052e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 35, time 680.14 s: f = -0.667785707694, ‖∇f‖ = 1.7863e-02, α = 5.54e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 36, time 689.37 s: f = -0.667885903685, ‖∇f‖ = 9.0771e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 37, time 698.32 s: f = -0.667941746884, ‖∇f‖ = 7.0177e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 38, time 708.15 s: f = -0.667974884585, ‖∇f‖ = 8.8422e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 39, time 716.81 s: f = -0.668048817152, ‖∇f‖ = 1.1324e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 40, time 725.46 s: f = -0.668123278247, ‖∇f‖ = 1.3967e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 41, time 735.36 s: f = -0.668188420240, ‖∇f‖ = 5.1732e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 42, time 744.40 s: f = -0.668212525320, ‖∇f‖ = 5.1150e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 43, time 754.13 s: f = -0.668245117146, ‖∇f‖ = 5.9778e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 44, time 763.24 s: f = -0.668321415567, ‖∇f‖ = 7.7043e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 45, time 781.85 s: f = -0.668349419497, ‖∇f‖ = 8.7093e-03, α = 3.23e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 46, time 790.85 s: f = -0.668394232920, ‖∇f‖ = 4.9490e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 47, time 799.85 s: f = -0.668423902729, ‖∇f‖ = 4.4315e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 48, time 809.56 s: f = -0.668453519413, ‖∇f‖ = 6.6590e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 49, time 818.73 s: f = -0.668482512627, ‖∇f‖ = 4.7810e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 50, time 827.88 s: f = -0.668522432273, ‖∇f‖ = 4.7205e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 51, time 842.27 s: f = -0.668536269464, ‖∇f‖ = 8.2957e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 52, time 851.35 s: f = -0.668554701367, ‖∇f‖ = 3.4938e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 53, time 860.86 s: f = -0.668564624860, ‖∇f‖ = 3.0996e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 54, time 870.08 s: f = -0.668580402016, ‖∇f‖ = 4.3886e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 55, time 879.06 s: f = -0.668603907003, ‖∇f‖ = 5.3988e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 56, time 897.98 s: f = -0.668617246170, ‖∇f‖ = 5.8694e-03, α = 4.99e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 57, time 907.72 s: f = -0.668632724020, ‖∇f‖ = 3.2358e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 58, time 917.01 s: f = -0.668645126493, ‖∇f‖ = 3.1763e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 59, time 926.33 s: f = -0.668655398082, ‖∇f‖ = 4.2974e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 60, time 936.00 s: f = -0.668669899756, ‖∇f‖ = 4.4860e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 61, time 945.03 s: f = -0.668685368558, ‖∇f‖ = 3.1222e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 62, time 954.24 s: f = -0.668694965356, ‖∇f‖ = 4.0039e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 63, time 963.47 s: f = -0.668703207321, ‖∇f‖ = 3.3668e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 64, time 972.78 s: f = -0.668712392397, ‖∇f‖ = 3.4980e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 65, time 981.96 s: f = -0.668728646271, ‖∇f‖ = 6.7104e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 66, time 991.59 s: f = -0.668739676361, ‖∇f‖ = 3.7165e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 67, time 1000.90 s: f = -0.668745430589, ‖∇f‖ = 2.1495e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 68, time 1010.57 s: f = -0.668751516160, ‖∇f‖ = 2.0809e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 69, time 1019.38 s: f = -0.668760790802, ‖∇f‖ = 2.5898e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 70, time 1028.31 s: f = -0.668768119147, ‖∇f‖ = 7.2679e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 71, time 1038.11 s: f = -0.668784264566, ‖∇f‖ = 2.6529e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 72, time 1047.35 s: f = -0.668789170375, ‖∇f‖ = 1.8137e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 73, time 1056.72 s: f = -0.668795881546, ‖∇f‖ = 2.0901e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 74, time 1066.35 s: f = -0.668799081816, ‖∇f‖ = 6.5956e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 75, time 1075.63 s: f = -0.668808063181, ‖∇f‖ = 2.8612e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 76, time 1085.30 s: f = -0.668814364081, ‖∇f‖ = 1.5758e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 77, time 1094.40 s: f = -0.668817946829, ‖∇f‖ = 1.9178e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 78, time 1103.72 s: f = -0.668824278385, ‖∇f‖ = 2.2613e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 79, time 1113.67 s: f = -0.668828126813, ‖∇f‖ = 4.6442e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 80, time 1123.07 s: f = -0.668835065826, ‖∇f‖ = 1.5939e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 81, time 1132.41 s: f = -0.668837215645, ‖∇f‖ = 2.3238e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 82, time 1142.13 s: f = -0.668837926114, ‖∇f‖ = 2.8015e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 83, time 1151.39 s: f = -0.668842788932, ‖∇f‖ = 1.8642e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 84, time 1160.72 s: f = -0.668846748527, ‖∇f‖ = 1.3959e-03, α = 1.00e+00, m = 20, nfg = 1
┌ Warning: LBFGS: not converged to requested tol after 85 iterations and time 1170.53 s: f = -0.668849701358, ‖∇f‖ = 2.0170e-03
└ @ OptimKit ~/.julia/packages/OptimKit/G6i79/src/lbfgs.jl:197
E = -0.6688497013575809
Note that for the specified parameters $J = \Delta = 1$, we simulated the same Hamiltonian as in the Heisenberg example. In that example, with a non-symmetric $D=2$ PEPS simulation, we reached a ground-state energy of around $E_\text{D=2} = -0.6625\dots$. Again comparing against Sandvik's accurate QMC estimate $E_{\text{ref}}=−0.6694421$, we see that we already got closer to the reference energy.
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