Optimizing the $U(1)$-symmetric Bose-Hubbard model
This example demonstrates the simulation of the two-dimensional Bose-Hubbard model. In particular, the point will be to showcase the use of internal symmetries and finite particle densities in PEPS ground state searches. As we will see, incorporating symmetries into the simulation consists of initializing a symmetric Hamiltonian, PEPS state and CTM environment - made possible through TensorKit.
But first let's seed the RNG and import the required modules:
using Random
using TensorKit, PEPSKit
using MPSKit: add_physical_charge
Random.seed!(2928528935);Defining the model
We will construct the Bose-Hubbard model Hamiltonian through the bose_hubbard_model, function from MPSKitModels as reexported by PEPSKit. We'll simulate the model in its Mott-insulating phase where the ratio $U/t$ is large, since in this phase we expect the ground state to be well approximated by a PEPS with a manifest global $U(1)$ symmetry. Furthermore, we'll impose a cutoff at 2 bosons per site, set the chemical potential to zero and use a simple $1 \times 1$ unit cell:
t = 1.0
U = 30.0
cutoff = 2
mu = 0.0
lattice = InfiniteSquare(1, 1);Next, we impose an explicit global $U(1)$ symmetry as well as a fixed particle number density in our simulations. We can do this by setting the symmetry argument of the Hamiltonian constructor to U1Irrep and passing one as the particle number density keyword argument n:
symmetry = U1Irrep
n = 1
H = bose_hubbard_model(ComplexF64, symmetry, lattice; cutoff, t, U, n);Before we continue, it might be interesting to inspect the corresponding lattice physical spaces (which is here just a $1 \times 1$ matrix due to the single-site unit cell):
physical_spaces = physicalspace(H)1×1 Matrix{TensorKit.GradedSpace{TensorKitSectors.U1Irrep, TensorKit.SortedVectorDict{TensorKitSectors.U1Irrep, Int64}}}:
Rep[TensorKitSectors.U₁](0=>1, 1=>1, -1=>1)Note that the physical space contains $U(1)$ charges -1, 0 and +1. Indeed, imposing a particle number density of +1 corresponds to shifting the physical charges by -1 to 're-center' the physical charges around the desired density. When we do this with a cutoff of two bosons per site, i.e. starting from $U(1)$ charges 0, 1 and 2 on the physical level, we indeed get the observed charges.
Characterizing the virtual spaces
When running PEPS simulations with explicit internal symmetries, specifying the structure of the virtual spaces of the PEPS and its environment becomes a bit more involved. For the environment, one could in principle allow the virtual space to be chosen dynamically during the boundary contraction using CTMRG by using a truncation scheme that allows for this (e.g. using alg=:truncdim or alg=:truncbelow to truncate to a fixed total bond dimension or singular value cutoff respectively). For the PEPS virtual space however, the structure has to be specified before the optimization.
While there are a host of techniques to do this in an informed way (e.g. starting from a simple update result), here we just specify the virtual space manually. Since we're dealing with a model at unit filling our physical space only contains integer $U(1)$ irreps. Therefore, we'll build our PEPS and environment spaces using integer $U(1)$ irreps centered around the zero charge:
V_peps = U1Space(0 => 2, 1 => 1, -1 => 1)
V_env = U1Space(0 => 6, 1 => 4, -1 => 4, 2 => 2, -2 => 2);Finding the ground state
Having defined our Hamiltonian and spaces, it is just a matter of plugging this into the optimization framework in the usual way to find the ground state. So, we first specify all algorithms and their tolerances:
boundary_alg = (; tol = 1.0e-8, alg = :simultaneous, trscheme = (; alg = :fixedspace))
gradient_alg = (; tol = 1.0e-6, maxiter = 10, alg = :eigsolver, iterscheme = :diffgauge)
optimizer_alg = (; tol = 1.0e-4, alg = :lbfgs, maxiter = 150, ls_maxiter = 2, ls_maxfg = 2);Taking CTMRG gradients and optimizing symmetric tensors tends to be more problematic than with dense tensors. In particular, this means that one frequently needs to tweak the boundary_alg, gradient_alg and optimizer_alg settings. There rarely is a general-purpose set of settings which will always work, so instead one has to adjust the simulation settings for each specific application. For example, it might help to switch between the CTMRG flavors alg=:simultaneous and alg=:sequential to improve convergence. The evaluation of the CTMRG gradient can be instable, so there it is advised to try the different iterscheme=:diffgauge and iterscheme=:fixed schemes as well as different alg keywords. Of course the tolerances of the algorithms and their subalgorithms also have to be compatible. For more details on the available options, see the fixedpoint docstring.
Keep in mind that the PEPS is constructed from a unit cell of spaces, so we have to make a matrix of V_peps spaces:
virtual_spaces = fill(V_peps, size(lattice)...)
peps₀ = InfinitePEPS(randn, ComplexF64, physical_spaces, virtual_spaces)
env₀, = leading_boundary(CTMRGEnv(peps₀, V_env), peps₀; boundary_alg...);[ Info: CTMRG init: obj = +1.693461429863e+00 +8.390974048721e-02im err = 1.0000e+00
[ Info: CTMRG conv 19: obj = +1.181834754305e+01 -1.514596781954e-11im err = 3.6943031807e-09 time = 8.97 sec
And at last, we optimize (which might take a bit):
peps, env, E, info = fixedpoint(
H, peps₀, env₀; boundary_alg, gradient_alg, optimizer_alg, verbosity = 3
)
@show E;[ Info: LBFGS: initializing with f = 9.360531870693, ‖∇f‖ = 1.6954e+01
[ Info: LBFGS: iter 1, time 835.55 s: f = 0.114269686001, ‖∇f‖ = 6.0686e+00, α = 1.56e+02, m = 0, nfg = 7
[ Info: LBFGS: iter 2, time 867.07 s: f = 0.059480938115, ‖∇f‖ = 7.2206e+00, α = 5.27e-01, m = 1, nfg = 2
[ Info: LBFGS: iter 3, time 870.15 s: f = -0.046499452141, ‖∇f‖ = 1.6329e+00, α = 1.00e+00, m = 2, nfg = 1
[ Info: LBFGS: iter 4, time 873.04 s: f = -0.079703746750, ‖∇f‖ = 1.4901e+00, α = 1.00e+00, m = 3, nfg = 1
[ Info: LBFGS: iter 5, time 883.62 s: f = -0.125317852222, ‖∇f‖ = 3.2630e+00, α = 5.23e-01, m = 4, nfg = 3
[ Info: LBFGS: iter 6, time 886.73 s: f = -0.163554919176, ‖∇f‖ = 1.2781e+00, α = 1.00e+00, m = 5, nfg = 1
[ Info: LBFGS: iter 7, time 890.21 s: f = -0.193532735237, ‖∇f‖ = 9.6932e-01, α = 1.00e+00, m = 6, nfg = 1
[ Info: LBFGS: iter 8, time 895.90 s: f = -0.208656321338, ‖∇f‖ = 7.0028e-01, α = 1.68e-01, m = 7, nfg = 2
[ Info: LBFGS: iter 9, time 901.97 s: f = -0.220718433148, ‖∇f‖ = 4.3381e-01, α = 3.95e-01, m = 8, nfg = 2
[ Info: LBFGS: iter 10, time 904.78 s: f = -0.227817345456, ‖∇f‖ = 5.8993e-01, α = 1.00e+00, m = 9, nfg = 1
[ Info: LBFGS: iter 11, time 906.29 s: f = -0.235906486649, ‖∇f‖ = 5.2265e-01, α = 1.00e+00, m = 10, nfg = 1
[ Info: LBFGS: iter 12, time 907.72 s: f = -0.245544719160, ‖∇f‖ = 3.6462e-01, α = 1.00e+00, m = 11, nfg = 1
[ Info: LBFGS: iter 13, time 909.56 s: f = -0.251717239130, ‖∇f‖ = 3.3074e-01, α = 1.00e+00, m = 12, nfg = 1
[ Info: LBFGS: iter 14, time 910.74 s: f = -0.256869388562, ‖∇f‖ = 2.9129e-01, α = 1.00e+00, m = 13, nfg = 1
[ Info: LBFGS: iter 15, time 911.84 s: f = -0.265345632631, ‖∇f‖ = 2.3580e-01, α = 1.00e+00, m = 14, nfg = 1
[ Info: LBFGS: iter 16, time 912.79 s: f = -0.267397827108, ‖∇f‖ = 3.0098e-01, α = 1.00e+00, m = 15, nfg = 1
[ Info: LBFGS: iter 17, time 914.23 s: f = -0.268894232390, ‖∇f‖ = 1.1725e-01, α = 1.00e+00, m = 16, nfg = 1
[ Info: LBFGS: iter 18, time 915.16 s: f = -0.269501536205, ‖∇f‖ = 8.8162e-02, α = 1.00e+00, m = 17, nfg = 1
[ Info: LBFGS: iter 19, time 916.08 s: f = -0.270154405721, ‖∇f‖ = 7.1880e-02, α = 1.00e+00, m = 18, nfg = 1
[ Info: LBFGS: iter 20, time 917.04 s: f = -0.270612692532, ‖∇f‖ = 6.5906e-02, α = 1.00e+00, m = 19, nfg = 1
[ Info: LBFGS: iter 21, time 918.52 s: f = -0.270978612734, ‖∇f‖ = 6.8050e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 22, time 919.42 s: f = -0.271251104411, ‖∇f‖ = 4.7832e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 23, time 920.32 s: f = -0.271592065725, ‖∇f‖ = 5.2245e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 24, time 921.24 s: f = -0.271907473230, ‖∇f‖ = 4.7783e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 25, time 922.68 s: f = -0.272188790518, ‖∇f‖ = 6.1727e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 26, time 923.58 s: f = -0.272341714938, ‖∇f‖ = 2.8588e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 27, time 924.50 s: f = -0.272416985001, ‖∇f‖ = 2.4404e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 28, time 925.41 s: f = -0.272488140123, ‖∇f‖ = 2.8167e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 29, time 926.85 s: f = -0.272607173823, ‖∇f‖ = 4.0551e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 30, time 927.74 s: f = -0.272669542118, ‖∇f‖ = 2.8338e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 31, time 928.65 s: f = -0.272710735515, ‖∇f‖ = 1.3171e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 32, time 929.56 s: f = -0.272737399244, ‖∇f‖ = 1.5064e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 33, time 931.01 s: f = -0.272785529235, ‖∇f‖ = 2.2115e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 34, time 931.92 s: f = -0.272869320156, ‖∇f‖ = 2.7454e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 35, time 932.84 s: f = -0.272917746471, ‖∇f‖ = 4.3200e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 36, time 933.76 s: f = -0.272982782958, ‖∇f‖ = 1.3998e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 37, time 935.17 s: f = -0.273001975617, ‖∇f‖ = 9.8876e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 38, time 936.04 s: f = -0.273014701175, ‖∇f‖ = 1.2336e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 39, time 936.94 s: f = -0.273032513148, ‖∇f‖ = 1.6628e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 40, time 937.86 s: f = -0.273047957579, ‖∇f‖ = 1.1548e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 41, time 939.28 s: f = -0.273056319317, ‖∇f‖ = 6.3307e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 42, time 940.16 s: f = -0.273062571624, ‖∇f‖ = 6.8219e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 43, time 941.04 s: f = -0.273067065008, ‖∇f‖ = 8.8510e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 44, time 941.96 s: f = -0.273077210295, ‖∇f‖ = 9.9163e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 45, time 944.28 s: f = -0.273086893655, ‖∇f‖ = 1.8575e-02, α = 5.12e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 46, time 945.20 s: f = -0.273103078820, ‖∇f‖ = 8.5718e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 47, time 946.11 s: f = -0.273110799145, ‖∇f‖ = 5.8581e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 48, time 947.54 s: f = -0.273120104689, ‖∇f‖ = 8.0410e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 49, time 948.44 s: f = -0.273131281118, ‖∇f‖ = 1.1878e-02, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 50, time 949.35 s: f = -0.273143808372, ‖∇f‖ = 9.4250e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 51, time 950.28 s: f = -0.273153887747, ‖∇f‖ = 7.2150e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 52, time 951.68 s: f = -0.273158885697, ‖∇f‖ = 6.7195e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 53, time 952.56 s: f = -0.273161233672, ‖∇f‖ = 4.1608e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 54, time 953.46 s: f = -0.273163225686, ‖∇f‖ = 4.0591e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 55, time 954.35 s: f = -0.273166294475, ‖∇f‖ = 4.9791e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 56, time 955.75 s: f = -0.273169366216, ‖∇f‖ = 4.4714e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 57, time 956.64 s: f = -0.273172354203, ‖∇f‖ = 6.3685e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 58, time 957.53 s: f = -0.273175363803, ‖∇f‖ = 3.9908e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 59, time 958.43 s: f = -0.273177279555, ‖∇f‖ = 3.9256e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 60, time 959.83 s: f = -0.273182789738, ‖∇f‖ = 6.6561e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 61, time 961.64 s: f = -0.273184790994, ‖∇f‖ = 5.6288e-03, α = 5.40e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 62, time 962.54 s: f = -0.273186538838, ‖∇f‖ = 2.7232e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 63, time 963.95 s: f = -0.273187761549, ‖∇f‖ = 2.8879e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 64, time 964.83 s: f = -0.273189383930, ‖∇f‖ = 3.5846e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 65, time 965.74 s: f = -0.273193896466, ‖∇f‖ = 8.4511e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 66, time 966.67 s: f = -0.273197826544, ‖∇f‖ = 6.5550e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 67, time 968.12 s: f = -0.273200889068, ‖∇f‖ = 3.7358e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 68, time 969.01 s: f = -0.273203155185, ‖∇f‖ = 3.5407e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 69, time 969.90 s: f = -0.273203929982, ‖∇f‖ = 3.6210e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 70, time 970.80 s: f = -0.273204838272, ‖∇f‖ = 3.2225e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 71, time 972.23 s: f = -0.273208065807, ‖∇f‖ = 3.2081e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 72, time 974.02 s: f = -0.273208664671, ‖∇f‖ = 2.6363e-03, α = 3.31e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 73, time 974.93 s: f = -0.273209100927, ‖∇f‖ = 2.4085e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 74, time 976.34 s: f = -0.273212054415, ‖∇f‖ = 3.6766e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 75, time 977.24 s: f = -0.273214539457, ‖∇f‖ = 4.4246e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 76, time 979.08 s: f = -0.273216162290, ‖∇f‖ = 4.5551e-03, α = 4.50e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 77, time 980.50 s: f = -0.273217804190, ‖∇f‖ = 2.2785e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 78, time 981.39 s: f = -0.273218883441, ‖∇f‖ = 2.0806e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 79, time 982.29 s: f = -0.273220350356, ‖∇f‖ = 2.6261e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 80, time 983.22 s: f = -0.273220936494, ‖∇f‖ = 6.7764e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 81, time 984.67 s: f = -0.273223126428, ‖∇f‖ = 2.3171e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 82, time 985.56 s: f = -0.273223758728, ‖∇f‖ = 1.4729e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 83, time 986.46 s: f = -0.273224312383, ‖∇f‖ = 1.6490e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 84, time 987.36 s: f = -0.273224595281, ‖∇f‖ = 5.4564e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 85, time 988.77 s: f = -0.273225489255, ‖∇f‖ = 2.6429e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 86, time 989.66 s: f = -0.273226533423, ‖∇f‖ = 1.3412e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 87, time 990.56 s: f = -0.273227338626, ‖∇f‖ = 1.9615e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 88, time 991.47 s: f = -0.273228071304, ‖∇f‖ = 2.3684e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 89, time 992.90 s: f = -0.273228783978, ‖∇f‖ = 2.1046e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 90, time 993.79 s: f = -0.273229430170, ‖∇f‖ = 1.6761e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 91, time 994.71 s: f = -0.273230395023, ‖∇f‖ = 2.3630e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 92, time 995.62 s: f = -0.273230913396, ‖∇f‖ = 4.6952e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 93, time 997.05 s: f = -0.273231822793, ‖∇f‖ = 2.4311e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 94, time 997.94 s: f = -0.273233027133, ‖∇f‖ = 2.0551e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 95, time 998.86 s: f = -0.273234061771, ‖∇f‖ = 3.0019e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 96, time 999.77 s: f = -0.273235399472, ‖∇f‖ = 3.6034e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 97, time 1001.23 s: f = -0.273236063107, ‖∇f‖ = 3.5368e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 98, time 1002.12 s: f = -0.273236795508, ‖∇f‖ = 1.2966e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 99, time 1003.01 s: f = -0.273237171417, ‖∇f‖ = 1.7109e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 100, time 1003.93 s: f = -0.273237667965, ‖∇f‖ = 2.2989e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 101, time 1006.24 s: f = -0.273237900160, ‖∇f‖ = 2.7773e-03, α = 4.53e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 102, time 1007.14 s: f = -0.273238275586, ‖∇f‖ = 1.4700e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 103, time 1008.06 s: f = -0.273238523826, ‖∇f‖ = 8.5901e-04, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 104, time 1009.47 s: f = -0.273238752012, ‖∇f‖ = 1.6628e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 105, time 1010.37 s: f = -0.273239111781, ‖∇f‖ = 2.6098e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 106, time 1011.27 s: f = -0.273239819089, ‖∇f‖ = 3.4353e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 107, time 1012.19 s: f = -0.273240369600, ‖∇f‖ = 4.2704e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 108, time 1013.60 s: f = -0.273241162430, ‖∇f‖ = 1.8052e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 109, time 1014.50 s: f = -0.273241475226, ‖∇f‖ = 1.0380e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 110, time 1015.42 s: f = -0.273241711280, ‖∇f‖ = 1.6135e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 111, time 1016.34 s: f = -0.273241988851, ‖∇f‖ = 2.1980e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 112, time 1017.75 s: f = -0.273242526227, ‖∇f‖ = 2.4062e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 113, time 1019.58 s: f = -0.273242706890, ‖∇f‖ = 2.5536e-03, α = 2.41e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 114, time 1020.49 s: f = -0.273243063849, ‖∇f‖ = 1.2275e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 115, time 1021.91 s: f = -0.273243259716, ‖∇f‖ = 8.8173e-04, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 116, time 1022.80 s: f = -0.273243390541, ‖∇f‖ = 1.2958e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 117, time 1023.71 s: f = -0.273243624187, ‖∇f‖ = 1.6522e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 118, time 1024.65 s: f = -0.273243975601, ‖∇f‖ = 2.4195e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 119, time 1026.08 s: f = -0.273244470238, ‖∇f‖ = 1.8095e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 120, time 1026.98 s: f = -0.273244906935, ‖∇f‖ = 1.1191e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 121, time 1027.88 s: f = -0.273245198892, ‖∇f‖ = 1.8385e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 122, time 1028.78 s: f = -0.273245436436, ‖∇f‖ = 1.7034e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 123, time 1030.25 s: f = -0.273245814839, ‖∇f‖ = 1.9152e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 124, time 1031.16 s: f = -0.273246901974, ‖∇f‖ = 2.4769e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 125, time 1032.06 s: f = -0.273247151000, ‖∇f‖ = 4.6605e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 126, time 1032.99 s: f = -0.273247895600, ‖∇f‖ = 1.6220e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 127, time 1034.43 s: f = -0.273248199597, ‖∇f‖ = 1.2653e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 128, time 1035.35 s: f = -0.273248721139, ‖∇f‖ = 2.0361e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 129, time 1036.28 s: f = -0.273249366483, ‖∇f‖ = 2.4923e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 130, time 1037.20 s: f = -0.273250035605, ‖∇f‖ = 3.6119e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 131, time 1038.61 s: f = -0.273250756926, ‖∇f‖ = 2.1636e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 132, time 1039.54 s: f = -0.273251085961, ‖∇f‖ = 1.1672e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 133, time 1040.45 s: f = -0.273251307038, ‖∇f‖ = 1.3676e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 134, time 1041.38 s: f = -0.273251484982, ‖∇f‖ = 1.6617e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 135, time 1042.81 s: f = -0.273251929354, ‖∇f‖ = 1.8732e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 136, time 1044.65 s: f = -0.273252100396, ‖∇f‖ = 1.8209e-03, α = 4.46e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 137, time 1045.58 s: f = -0.273252323246, ‖∇f‖ = 1.1334e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 138, time 1047.02 s: f = -0.273252503681, ‖∇f‖ = 1.3498e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 139, time 1047.93 s: f = -0.273252679906, ‖∇f‖ = 1.8133e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 140, time 1048.85 s: f = -0.273253041604, ‖∇f‖ = 1.9959e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 141, time 1049.78 s: f = -0.273253394913, ‖∇f‖ = 2.7713e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 142, time 1051.23 s: f = -0.273253983127, ‖∇f‖ = 1.4203e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 143, time 1052.13 s: f = -0.273254647402, ‖∇f‖ = 2.1096e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 144, time 1053.05 s: f = -0.273255259635, ‖∇f‖ = 2.8357e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 145, time 1053.99 s: f = -0.273256194806, ‖∇f‖ = 3.9979e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 146, time 1056.37 s: f = -0.273256707532, ‖∇f‖ = 2.3435e-03, α = 4.40e-01, m = 20, nfg = 2
[ Info: LBFGS: iter 147, time 1057.29 s: f = -0.273257209926, ‖∇f‖ = 1.0557e-03, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 148, time 1058.22 s: f = -0.273257438291, ‖∇f‖ = 9.7121e-04, α = 1.00e+00, m = 20, nfg = 1
[ Info: LBFGS: iter 149, time 1059.67 s: f = -0.273257563053, ‖∇f‖ = 1.1509e-03, α = 1.00e+00, m = 20, nfg = 1
┌ Warning: LBFGS: not converged to requested tol after 150 iterations and time 1060.58 s: f = -0.273257823816, ‖∇f‖ = 1.5465e-03
└ @ OptimKit ~/.julia/packages/OptimKit/G6i79/src/lbfgs.jl:197
E = -0.2732578238158397
We can compare our PEPS result to the energy obtained using a cylinder-MPS calculation using a cylinder circumference of $L_y = 7$ and a bond dimension of 446, which yields $E = -0.273284888$:
E_ref = -0.273284888
@show (E - E_ref) / E_ref;(E - E_ref) / E_ref = -9.90328603911651e-5
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