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 = 1.0e-8, alg = :simultaneous, trscheme = (; alg = :fixedspace))
gradient_alg = (; tol = 1.0e-6, alg = :eigsolver, maxiter = 10, iterscheme = :diffgauge)
optimizer_alg = (; tol = 1.0e-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 -4.009734766441e-08im err = 5.3638613489e-09 time = 3.81 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 106.06 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 181.45 s: f = -0.527654857567, ‖∇f‖ = 7.5002e-01, α = 2.50e+01, m = 0, nfg = 4
[ Info: LBFGS: iter 3, time 199.44 s: f = -0.552751928887, ‖∇f‖ = 3.6310e-01, α = 1.00e+00, m = 1, nfg = 1
[ Info: LBFGS: iter 4, time 253.43 s: f = -0.617206129929, ‖∇f‖ = 3.0891e-01, α = 3.20e+00, m = 2, nfg = 3
[ Info: LBFGS: iter 5, time 272.00 s: f = -0.632819487274, ‖∇f‖ = 4.0073e-01, α = 1.00e+00, m = 3, nfg = 1
[ Info: LBFGS: iter 6, time 288.78 s: f = -0.653138513637, ‖∇f‖ = 1.0717e-01, α = 1.00e+00, m = 4, nfg = 1
[ Info: LBFGS: iter 7, time 298.89 s: f = -0.655569926077, ‖∇f‖ = 4.7861e-02, α = 1.00e+00, m = 5, nfg = 1
[ Info: LBFGS: iter 8, time 310.29 s: f = -0.656621018321, ‖∇f‖ = 4.3322e-02, α = 1.00e+00, m = 6, nfg = 1
[ Info: LBFGS: iter 9, time 320.40 s: f = -0.657996751278, ‖∇f‖ = 4.6277e-02, α = 1.00e+00, m = 7, nfg = 1
[ Info: LBFGS: iter 10, time 329.76 s: f = -0.659837382098, ‖∇f‖ = 4.8930e-02, α = 1.00e+00, m = 8, nfg = 1
[ Info: LBFGS: iter 11, time 340.25 s: f = -0.661058228215, ‖∇f‖ = 4.1864e-02, α = 1.00e+00, m = 9, nfg = 1
[ Info: LBFGS: iter 12, time 349.38 s: f = -0.661633890101, ‖∇f‖ = 1.7067e-02, α = 1.00e+00, m = 10, nfg = 1
[ Info: LBFGS: iter 13, time 357.68 s: f = -0.661839831812, ‖∇f‖ = 1.4709e-02, α = 1.00e+00, m = 11, nfg = 1
[ Info: LBFGS: iter 14, time 367.25 s: f = -0.662134636143, ‖∇f‖ = 1.7065e-02, α = 1.00e+00, m = 12, nfg = 1
[ Info: LBFGS: iter 15, time 375.64 s: f = -0.662532716659, ‖∇f‖ = 1.7719e-02, α = 1.00e+00, m = 13, nfg = 1
[ Info: LBFGS: iter 16, time 383.89 s: f = -0.662996909612, ‖∇f‖ = 2.0543e-02, α = 1.00e+00, m = 14, nfg = 1
[ Info: LBFGS: iter 17, time 393.35 s: f = -0.663439633406, ‖∇f‖ = 2.6049e-02, α = 1.00e+00, m = 15, nfg = 1
[ Info: LBFGS: iter 18, time 401.54 s: f = -0.663855589525, ‖∇f‖ = 3.2567e-02, α = 1.00e+00, m = 16, nfg = 1
[ Info: LBFGS: iter 19, time 409.75 s: f = -0.664450880841, ‖∇f‖ = 2.3691e-02, α = 1.00e+00, m = 17, nfg = 1
[ Info: LBFGS: iter 20, time 419.86 s: f = -0.664917088518, ‖∇f‖ = 1.8281e-02, α = 1.00e+00, m = 18, nfg = 1
[ Info: LBFGS: iter 21, time 428.21 s: f = -0.665079077676, ‖∇f‖ = 1.2380e-02, α = 1.00e+00, m = 19, nfg = 1
[ Info: LBFGS: iter 22, time 436.46 s: f = -0.665287463007, ‖∇f‖ = 1.7237e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 23, time 445.88 s: f = -0.665717355832, ‖∇f‖ = 2.9356e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 24, time 454.89 s: f = -0.666166129443, ‖∇f‖ = 3.2946e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 25, time 474.33 s: f = -0.666407108755, ‖∇f‖ = 2.6211e-02, α = 4.29e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 26, time 483.35 s: f = -0.666612715340, ‖∇f‖ = 1.1288e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 27, time 492.43 s: f = -0.666709056953, ‖∇f‖ = 1.2143e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 28, time 502.05 s: f = -0.666869683924, ‖∇f‖ = 1.8077e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 29, time 511.17 s: f = -0.667130274911, ‖∇f‖ = 2.1430e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 30, time 530.62 s: f = -0.667250996806, ‖∇f‖ = 2.4838e-02, α = 4.17e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 31, time 539.67 s: f = -0.667422880816, ‖∇f‖ = 1.0123e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 32, time 548.64 s: f = -0.667499816879, ‖∇f‖ = 7.8017e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 33, time 558.89 s: f = -0.667576990274, ‖∇f‖ = 1.2346e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 34, time 568.08 s: f = -0.667710803542, ‖∇f‖ = 1.5052e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 35, time 586.23 s: f = -0.667785707694, ‖∇f‖ = 1.7863e-02, α = 5.54e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 36, time 594.68 s: f = -0.667885903685, ‖∇f‖ = 9.0771e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 37, time 603.14 s: f = -0.667941746884, ‖∇f‖ = 7.0177e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 38, time 612.88 s: f = -0.667974884585, ‖∇f‖ = 8.8422e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 39, time 621.44 s: f = -0.668048817152, ‖∇f‖ = 1.1324e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 40, time 630.01 s: f = -0.668123278247, ‖∇f‖ = 1.3967e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 41, time 639.72 s: f = -0.668188420240, ‖∇f‖ = 5.1732e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 42, time 648.20 s: f = -0.668212525320, ‖∇f‖ = 5.1150e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 43, time 656.80 s: f = -0.668245117146, ‖∇f‖ = 5.9778e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 44, time 666.45 s: f = -0.668321415567, ‖∇f‖ = 7.7043e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 45, time 683.91 s: f = -0.668349419497, ‖∇f‖ = 8.7093e-03, α = 3.23e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 46, time 693.71 s: f = -0.668394232920, ‖∇f‖ = 4.9490e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 47, time 702.37 s: f = -0.668423902729, ‖∇f‖ = 4.4315e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 48, time 711.06 s: f = -0.668453519414, ‖∇f‖ = 6.6590e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 49, time 720.88 s: f = -0.668482512627, ‖∇f‖ = 4.7810e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 50, time 729.57 s: f = -0.668522432273, ‖∇f‖ = 4.7205e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 51, time 744.60 s: f = -0.668536269464, ‖∇f‖ = 8.2957e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 52, time 754.55 s: f = -0.668554701367, ‖∇f‖ = 3.4938e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 53, time 763.10 s: f = -0.668564624860, ‖∇f‖ = 3.0996e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 54, time 771.72 s: f = -0.668580402015, ‖∇f‖ = 4.3886e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 55, time 781.60 s: f = -0.668603907002, ‖∇f‖ = 5.3988e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 56, time 799.28 s: f = -0.668617246170, ‖∇f‖ = 5.8694e-03, α = 4.99e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 57, time 809.22 s: f = -0.668632724020, ‖∇f‖ = 3.2358e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 58, time 817.86 s: f = -0.668645126493, ‖∇f‖ = 3.1763e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 59, time 826.48 s: f = -0.668655398082, ‖∇f‖ = 4.2974e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 60, time 836.41 s: f = -0.668669899756, ‖∇f‖ = 4.4860e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 61, time 845.29 s: f = -0.668685368558, ‖∇f‖ = 3.1222e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 62, time 854.05 s: f = -0.668694965356, ‖∇f‖ = 4.0039e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 63, time 864.05 s: f = -0.668703207322, ‖∇f‖ = 3.3668e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 64, time 872.74 s: f = -0.668712392397, ‖∇f‖ = 3.4980e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 65, time 881.52 s: f = -0.668728646277, ‖∇f‖ = 6.7104e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 66, time 891.65 s: f = -0.668739676358, ‖∇f‖ = 3.7164e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 67, time 900.41 s: f = -0.668745430593, ‖∇f‖ = 2.1495e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 68, time 909.04 s: f = -0.668751516168, ‖∇f‖ = 2.0809e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 69, time 918.90 s: f = -0.668760790816, ‖∇f‖ = 2.5898e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 70, time 927.79 s: f = -0.668768119255, ‖∇f‖ = 7.2679e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 71, time 936.79 s: f = -0.668784264577, ‖∇f‖ = 2.6529e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 72, time 946.72 s: f = -0.668789170407, ‖∇f‖ = 1.8137e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 73, time 955.52 s: f = -0.668795881540, ‖∇f‖ = 2.0901e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 74, time 964.44 s: f = -0.668799082134, ‖∇f‖ = 6.5955e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 75, time 974.47 s: f = -0.668808063314, ‖∇f‖ = 2.8612e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 76, time 983.39 s: f = -0.668814364357, ‖∇f‖ = 1.5758e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 77, time 992.33 s: f = -0.668817946985, ‖∇f‖ = 1.9177e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 78, time 1002.50 s: f = -0.668824278695, ‖∇f‖ = 2.2613e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 79, time 1011.45 s: f = -0.668828127692, ‖∇f‖ = 4.6437e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 80, time 1020.26 s: f = -0.668835066382, ‖∇f‖ = 1.5939e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 81, time 1030.51 s: f = -0.668837216668, ‖∇f‖ = 2.3232e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 82, time 1039.35 s: f = -0.668837924470, ‖∇f‖ = 2.8024e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 83, time 1048.22 s: f = -0.668842790098, ‖∇f‖ = 1.8645e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 84, time 1058.49 s: f = -0.668846749446, ‖∇f‖ = 1.3957e-03, α = 1.00e+00, m = 20, nfg = 1
┌ Warning: LBFGS: not converged to requested tol after 85 iterations and time 1067.39 s: f = -0.668849701737, ‖∇f‖ = 2.0164e-03
└ @ OptimKit ~/.julia/packages/OptimKit/G6i79/src/lbfgs.jl:197
E = -0.668849701737108
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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