The XXZ model
In this file we will give step by step instructions on how to analyze the spin 1/2 XXZ model. The necessary packages to follow this tutorial are:
using MPSKit, MPSKitModels, TensorKit, PlotsFailure
First we should define the hamiltonian we want to work with. Then we specify an initial guess, which we then further optimize. Working directly in the thermodynamic limit, this is achieved as follows:
H = heisenberg_XXX(; spin=1 // 2)single site InfiniteMPOHamiltonian{BlockTensorKit.SparseBlockTensorMap{TensorKit.AbstractTensorMap{ComplexF64, TensorKit.ComplexSpace, 2, 2}, ComplexF64, TensorKit.ComplexSpace, 2, 2, 4}}:
╷ ⋮
┼ W[1]: 3×1×1×3 SparseBlockTensorMap(((ℂ^1 ⊕ ℂ^3 ⊕ ℂ^1) ⊗ ⊕(ℂ^2)) ← (⊕(ℂ^2) ⊗ (ℂ^1 ⊕ ℂ^3 ⊕ ℂ^1)))
╵ ⋮
We then need an intial state, which we shall later optimize. In this example we work directly in the thermodynamic limit.
state = InfiniteMPS(2, 20)single site InfiniteMPS:
│ ⋮
│ C[1]: TensorMap(ℂ^20 ← ℂ^20)
├── AL[1]: TensorMap((ℂ^20 ⊗ ℂ^2) ← ℂ^20)
│ ⋮
The groundstate can then be found by calling find_groundstate.
groundstate, cache, delta = find_groundstate(state, H, VUMPS());[ Info: VUMPS init: obj = +2.499932270895e-01 err = 7.3488e-03
[ Info: VUMPS 1: obj = -2.684294662883e-01 err = 3.4887991618e-01 time = 0.04 sec
[ Info: VUMPS 2: obj = -3.908732509075e-01 err = 2.4346308901e-01 time = 0.09 sec
[ Info: VUMPS 3: obj = -4.103272462718e-01 err = 2.2211707874e-01 time = 0.04 sec
[ Info: VUMPS 4: obj = +2.763771623319e-02 err = 4.0601070717e-01 time = 0.06 sec
[ Info: VUMPS 5: obj = -1.278912388343e-01 err = 4.0024753668e-01 time = 0.04 sec
[ Info: VUMPS 6: obj = -6.128684560655e-02 err = 3.9554377744e-01 time = 0.05 sec
[ Info: VUMPS 7: obj = -2.019229524257e-01 err = 3.9507954388e-01 time = 0.03 sec
[ Info: VUMPS 8: obj = -3.385836978728e-01 err = 3.1206810658e-01 time = 0.06 sec
[ Info: VUMPS 9: obj = -4.217712628117e-01 err = 1.9266010859e-01 time = 0.05 sec
[ Info: VUMPS 10: obj = +3.119439956568e-02 err = 4.0307569557e-01 time = 0.05 sec
[ Info: VUMPS 11: obj = -1.453987450329e-01 err = 3.7892383130e-01 time = 0.03 sec
[ Info: VUMPS 12: obj = -3.122541983387e-01 err = 3.4361928372e-01 time = 0.05 sec
[ Info: VUMPS 13: obj = -3.175415058698e-01 err = 3.4372788072e-01 time = 0.06 sec
[ Info: VUMPS 14: obj = -9.281281368249e-02 err = 3.8082690832e-01 time = 0.04 sec
[ Info: VUMPS 15: obj = -1.118289260232e-01 err = 3.9703529501e-01 time = 0.04 sec
[ Info: VUMPS 16: obj = -3.392299177529e-02 err = 3.8653278445e-01 time = 0.04 sec
[ Info: VUMPS 17: obj = -8.859386904872e-02 err = 3.7559265639e-01 time = 0.06 sec
[ Info: VUMPS 18: obj = -2.660652072169e-01 err = 3.4535690909e-01 time = 0.06 sec
[ Info: VUMPS 19: obj = +1.581362017845e-01 err = 3.6728199421e-01 time = 0.02 sec
[ Info: VUMPS 20: obj = -2.614275413781e-03 err = 3.5210479173e-01 time = 0.04 sec
[ Info: VUMPS 21: obj = -1.404557974268e-01 err = 3.8727147114e-01 time = 0.02 sec
[ Info: VUMPS 22: obj = -1.922542821356e-01 err = 3.6768505428e-01 time = 0.05 sec
[ Info: VUMPS 23: obj = +8.174391358782e-02 err = 3.9158252120e-01 time = 0.02 sec
[ Info: VUMPS 24: obj = +4.347184217158e-02 err = 3.3940461711e-01 time = 0.05 sec
[ Info: VUMPS 25: obj = -2.122928828839e-01 err = 3.6816674483e-01 time = 0.02 sec
[ Info: VUMPS 26: obj = -3.750381325917e-02 err = 4.4510656286e-01 time = 0.06 sec
[ Info: VUMPS 27: obj = -9.698985317742e-02 err = 4.0266067497e-01 time = 0.05 sec
[ Info: VUMPS 28: obj = -2.160914808205e-01 err = 3.6590508687e-01 time = 0.03 sec
[ Info: VUMPS 29: obj = +4.936490579149e-02 err = 3.7234044270e-01 time = 0.05 sec
[ Info: VUMPS 30: obj = -1.148676233851e-01 err = 4.1462641635e-01 time = 0.02 sec
[ Info: VUMPS 31: obj = -1.519618595135e-01 err = 3.7492975708e-01 time = 0.05 sec
[ Info: VUMPS 32: obj = -1.890862522082e-01 err = 3.7659388392e-01 time = 0.03 sec
[ Info: VUMPS 33: obj = -7.477276890967e-02 err = 3.9194272124e-01 time = 0.05 sec
[ Info: VUMPS 34: obj = -1.940989007410e-01 err = 3.9403330192e-01 time = 0.03 sec
[ Info: VUMPS 35: obj = -3.502934869819e-01 err = 3.0111969128e-01 time = 0.05 sec
[ Info: VUMPS 36: obj = -3.442085541276e-01 err = 3.2611257359e-01 time = 0.04 sec
[ Info: VUMPS 37: obj = -2.312483202161e-01 err = 3.7396977836e-01 time = 0.06 sec
[ Info: VUMPS 38: obj = -2.282920202884e-01 err = 3.5801388290e-01 time = 0.06 sec
[ Info: VUMPS 39: obj = -3.434233746212e-01 err = 3.3005413456e-01 time = 0.05 sec
[ Info: VUMPS 40: obj = -3.700311228608e-01 err = 2.8698807789e-01 time = 0.05 sec
[ Info: VUMPS 41: obj = -2.287419146432e-01 err = 3.7652136991e-01 time = 0.08 sec
[ Info: VUMPS 42: obj = -1.848386897474e-01 err = 3.9410789712e-01 time = 0.07 sec
[ Info: VUMPS 43: obj = -4.151049475911e-02 err = 3.6614428192e-01 time = 0.04 sec
[ Info: VUMPS 44: obj = +4.278404791371e-02 err = 3.8999971963e-01 time = 0.06 sec
[ Info: VUMPS 45: obj = +4.133615949384e-02 err = 3.7259192609e-01 time = 0.04 sec
[ Info: VUMPS 46: obj = -1.755417704686e-01 err = 3.8281160487e-01 time = 0.06 sec
[ Info: VUMPS 47: obj = -3.433301511715e-01 err = 3.1797681990e-01 time = 0.03 sec
[ Info: VUMPS 48: obj = +8.038768523644e-02 err = 3.5190647487e-01 time = 0.05 sec
[ Info: VUMPS 49: obj = -6.209071290954e-02 err = 3.4368443495e-01 time = 0.07 sec
[ Info: VUMPS 50: obj = +1.373233695943e-01 err = 3.4584780766e-01 time = 0.04 sec
[ Info: VUMPS 51: obj = -1.298395809866e-02 err = 3.7394444037e-01 time = 0.06 sec
[ Info: VUMPS 52: obj = -1.849662152303e-01 err = 4.0186739319e-01 time = 0.04 sec
[ Info: VUMPS 53: obj = -2.535592561522e-01 err = 3.6315928824e-01 time = 0.05 sec
[ Info: VUMPS 54: obj = -2.078073506292e-01 err = 3.8862703529e-01 time = 0.03 sec
[ Info: VUMPS 55: obj = -2.648807481312e-01 err = 3.5935081624e-01 time = 0.05 sec
[ Info: VUMPS 56: obj = +3.508321588679e-02 err = 3.5155300157e-01 time = 0.06 sec
[ Info: VUMPS 57: obj = +2.587094014595e-02 err = 3.7426723561e-01 time = 0.05 sec
[ Info: VUMPS 58: obj = +6.099768411641e-02 err = 3.6261231989e-01 time = 0.06 sec
[ Info: VUMPS 59: obj = +6.085060881519e-02 err = 3.5718570541e-01 time = 0.02 sec
[ Info: VUMPS 60: obj = -4.856039331256e-02 err = 3.9347118222e-01 time = 0.06 sec
[ Info: VUMPS 61: obj = -5.072268826867e-02 err = 3.8271040330e-01 time = 0.04 sec
[ Info: VUMPS 62: obj = +1.769142598093e-02 err = 4.0183932016e-01 time = 0.05 sec
[ Info: VUMPS 63: obj = +5.913223876388e-02 err = 4.0964563195e-01 time = 0.06 sec
[ Info: VUMPS 64: obj = -2.478892754727e-01 err = 3.8032380630e-01 time = 0.04 sec
[ Info: VUMPS 65: obj = -3.071655369627e-01 err = 3.4522086404e-01 time = 0.05 sec
[ Info: VUMPS 66: obj = -3.204977019062e-01 err = 3.2905910740e-01 time = 0.05 sec
[ Info: VUMPS 67: obj = -2.546816866134e-01 err = 3.8561093133e-01 time = 0.07 sec
[ Info: VUMPS 68: obj = -2.393554685021e-01 err = 3.5880361408e-01 time = 0.05 sec
[ Info: VUMPS 69: obj = -1.327133696952e-01 err = 4.0101733932e-01 time = 0.06 sec
[ Info: VUMPS 70: obj = -1.041902383608e-01 err = 4.1285042628e-01 time = 0.05 sec
[ Info: VUMPS 71: obj = -1.993915263334e-01 err = 3.6003918734e-01 time = 0.04 sec
[ Info: VUMPS 72: obj = -1.940520154241e-01 err = 3.4570202542e-01 time = 0.02 sec
[ Info: VUMPS 73: obj = -8.071173596559e-02 err = 3.9919344455e-01 time = 0.04 sec
[ Info: VUMPS 74: obj = -1.370928589038e-01 err = 3.8043789146e-01 time = 0.02 sec
[ Info: VUMPS 75: obj = -2.590488646429e-01 err = 3.6030703598e-01 time = 0.05 sec
[ Info: VUMPS 76: obj = -4.074845294173e-01 err = 2.2747635384e-01 time = 0.03 sec
[ Info: VUMPS 77: obj = -4.083741658015e-01 err = 2.2604491079e-01 time = 0.05 sec
[ Info: VUMPS 78: obj = +1.005627309383e-01 err = 3.4879269264e-01 time = 0.05 sec
[ Info: VUMPS 79: obj = -1.534111629484e-01 err = 4.0398980845e-01 time = 0.04 sec
[ Info: VUMPS 80: obj = -1.185244700305e-01 err = 4.1527071676e-01 time = 0.05 sec
[ Info: VUMPS 81: obj = -2.203348020820e-01 err = 3.4143151318e-01 time = 0.04 sec
[ Info: VUMPS 82: obj = -5.889431350950e-02 err = 3.9822663932e-01 time = 0.05 sec
[ Info: VUMPS 83: obj = -1.737018741265e-01 err = 3.8538020144e-01 time = 0.04 sec
[ Info: VUMPS 84: obj = -1.582331643179e-01 err = 3.6506684092e-01 time = 0.02 sec
[ Info: VUMPS 85: obj = -2.020317867236e-01 err = 3.7807390527e-01 time = 0.05 sec
[ Info: VUMPS 86: obj = -6.233681285915e-02 err = 3.7406473722e-01 time = 0.02 sec
[ Info: VUMPS 87: obj = -1.314218634692e-01 err = 3.8818132852e-01 time = 0.06 sec
[ Info: VUMPS 88: obj = -2.302777507565e-01 err = 3.6235120973e-01 time = 0.04 sec
[ Info: VUMPS 89: obj = +1.777592295373e-02 err = 3.1886054992e-01 time = 0.06 sec
[ Info: VUMPS 90: obj = -9.137389133038e-02 err = 3.6650533950e-01 time = 0.05 sec
[ Info: VUMPS 91: obj = -1.522621453585e-01 err = 3.7520570789e-01 time = 0.04 sec
[ Info: VUMPS 92: obj = -2.067248068564e-01 err = 3.8667875904e-01 time = 0.05 sec
[ Info: VUMPS 93: obj = +2.047786471459e-02 err = 3.7528645946e-01 time = 0.04 sec
[ Info: VUMPS 94: obj = -2.196126737735e-01 err = 3.7361837770e-01 time = 0.06 sec
[ Info: VUMPS 95: obj = -3.148681836053e-01 err = 3.3877001456e-01 time = 0.03 sec
[ Info: VUMPS 96: obj = -3.632549896064e-01 err = 3.1063135089e-01 time = 0.05 sec
[ Info: VUMPS 97: obj = -2.359487524961e-01 err = 3.4983505936e-01 time = 0.05 sec
[ Info: VUMPS 98: obj = -2.085648662363e-01 err = 3.6646074907e-01 time = 0.04 sec
[ Info: VUMPS 99: obj = -3.378709599647e-01 err = 3.1090195525e-01 time = 0.03 sec
[ Info: VUMPS 100: obj = +1.697054331465e-01 err = 3.3318043865e-01 time = 0.03 sec
[ Info: VUMPS 101: obj = -1.372336728300e-01 err = 3.9831002380e-01 time = 0.03 sec
[ Info: VUMPS 102: obj = -2.025796302451e-01 err = 3.6198000343e-01 time = 0.05 sec
[ Info: VUMPS 103: obj = -2.078176354072e-01 err = 3.8150634396e-01 time = 0.03 sec
[ Info: VUMPS 104: obj = -2.923528623190e-01 err = 3.3199471479e-01 time = 0.09 sec
[ Info: VUMPS 105: obj = -2.895261890513e-02 err = 3.9381156810e-01 time = 0.06 sec
[ Info: VUMPS 106: obj = +1.102282311350e-01 err = 3.6753101837e-01 time = 0.02 sec
[ Info: VUMPS 107: obj = -3.457075224669e-03 err = 3.8935262670e-01 time = 0.05 sec
[ Info: VUMPS 108: obj = -1.401802303936e-01 err = 3.5477482436e-01 time = 0.07 sec
[ Info: VUMPS 109: obj = -2.773177515742e-01 err = 3.5618814609e-01 time = 0.04 sec
[ Info: VUMPS 110: obj = -2.980497929052e-01 err = 3.3973067915e-01 time = 0.07 sec
[ Info: VUMPS 111: obj = -3.666221447505e-01 err = 2.9612511089e-01 time = 0.04 sec
[ Info: VUMPS 112: obj = -3.406894034780e-01 err = 3.1838335674e-01 time = 0.07 sec
[ Info: VUMPS 113: obj = -4.117451756841e-01 err = 2.1229269280e-01 time = 0.05 sec
[ Info: VUMPS 114: obj = +3.472393021263e-02 err = 4.2138653263e-01 time = 0.04 sec
[ Info: VUMPS 115: obj = +6.929635845720e-03 err = 4.0042995319e-01 time = 0.02 sec
[ Info: VUMPS 116: obj = +1.198686632180e-01 err = 3.5017433469e-01 time = 0.04 sec
[ Info: VUMPS 117: obj = -1.271079129807e-01 err = 3.7101933225e-01 time = 0.02 sec
[ Info: VUMPS 118: obj = -1.834088729116e-01 err = 3.6678418731e-01 time = 0.03 sec
[ Info: VUMPS 119: obj = -3.363021880354e-01 err = 3.1503549193e-01 time = 0.02 sec
[ Info: VUMPS 120: obj = -4.275043892741e-01 err = 1.5610761856e-01 time = 0.06 sec
[ Info: VUMPS 121: obj = +3.978743301000e-02 err = 3.1230305054e-01 time = 0.08 sec
[ Info: VUMPS 122: obj = -1.990612171666e-01 err = 3.6200810172e-01 time = 0.04 sec
[ Info: VUMPS 123: obj = +1.050122944992e-01 err = 3.4198140893e-01 time = 0.04 sec
[ Info: VUMPS 124: obj = -5.398579365216e-02 err = 4.1624422030e-01 time = 0.04 sec
[ Info: VUMPS 125: obj = -1.003705443385e-01 err = 3.7636491379e-01 time = 0.02 sec
[ Info: VUMPS 126: obj = +8.454616496648e-02 err = 3.5803981901e-01 time = 0.04 sec
[ Info: VUMPS 127: obj = -1.128523326267e-01 err = 3.8181570938e-01 time = 0.03 sec
[ Info: VUMPS 128: obj = -2.150635416320e-01 err = 3.5780210657e-01 time = 0.04 sec
[ Info: VUMPS 129: obj = -1.312465126627e-01 err = 3.8405159732e-01 time = 0.06 sec
[ Info: VUMPS 130: obj = -3.846997189431e-02 err = 4.2623757858e-01 time = 0.05 sec
[ Info: VUMPS 131: obj = +2.473200047618e-02 err = 3.5905352543e-01 time = 0.05 sec
[ Info: VUMPS 132: obj = +7.006948564206e-02 err = 3.4845271565e-01 time = 0.04 sec
[ Info: VUMPS 133: obj = -2.383461944993e-01 err = 3.7052130221e-01 time = 0.06 sec
[ Info: VUMPS 134: obj = -2.062217898227e-01 err = 3.6937256423e-01 time = 0.04 sec
[ Info: VUMPS 135: obj = -2.274384652936e-01 err = 3.4724295244e-01 time = 0.07 sec
[ Info: VUMPS 136: obj = -2.622435856918e-01 err = 3.4143263011e-01 time = 0.05 sec
[ Info: VUMPS 137: obj = -1.368393705006e-01 err = 3.9505761386e-01 time = 0.04 sec
[ Info: VUMPS 138: obj = -2.365952891363e-01 err = 3.6070782003e-01 time = 0.05 sec
[ Info: VUMPS 139: obj = -3.285892419662e-01 err = 3.2510337702e-01 time = 0.06 sec
[ Info: VUMPS 140: obj = -2.897155176623e-01 err = 3.4971272209e-01 time = 0.05 sec
[ Info: VUMPS 141: obj = -5.613172601920e-02 err = 4.0280146205e-01 time = 0.06 sec
[ Info: VUMPS 142: obj = -1.912055510339e-01 err = 3.8086840086e-01 time = 0.06 sec
[ Info: VUMPS 143: obj = -2.522493416812e-01 err = 3.7780461889e-01 time = 0.03 sec
[ Info: VUMPS 144: obj = -1.448388260442e-01 err = 3.9735778286e-01 time = 0.06 sec
[ Info: VUMPS 145: obj = +2.133811571892e-02 err = 3.6767706542e-01 time = 0.05 sec
[ Info: VUMPS 146: obj = -1.021149430836e-01 err = 4.1518428940e-01 time = 0.04 sec
[ Info: VUMPS 147: obj = +3.345423157498e-02 err = 3.7487854814e-01 time = 0.06 sec
[ Info: VUMPS 148: obj = -1.519027769061e-01 err = 3.7323297896e-01 time = 0.04 sec
[ Info: VUMPS 149: obj = -1.015157315769e-01 err = 3.9037495750e-01 time = 0.07 sec
[ Info: VUMPS 150: obj = -7.525226965218e-02 err = 4.0412444789e-01 time = 0.05 sec
[ Info: VUMPS 151: obj = -1.435426260194e-01 err = 4.0959006551e-01 time = 0.08 sec
[ Info: VUMPS 152: obj = -1.234168209435e-01 err = 4.0252461632e-01 time = 0.02 sec
[ Info: VUMPS 153: obj = -1.582356991859e-01 err = 4.0856159774e-01 time = 0.07 sec
[ Info: VUMPS 154: obj = -2.611502661937e-02 err = 3.7361316068e-01 time = 0.02 sec
[ Info: VUMPS 155: obj = -1.903742563432e-02 err = 3.6268799414e-01 time = 0.05 sec
[ Info: VUMPS 156: obj = -1.660075702137e-01 err = 3.6001499475e-01 time = 0.04 sec
[ Info: VUMPS 157: obj = -1.750801644509e-01 err = 3.5799477869e-01 time = 0.03 sec
[ Info: VUMPS 158: obj = -3.185357562769e-01 err = 3.4039580564e-01 time = 0.04 sec
[ Info: VUMPS 159: obj = -3.462765635183e-01 err = 3.0750748489e-01 time = 0.03 sec
[ Info: VUMPS 160: obj = -3.906343752662e-01 err = 2.7025958372e-01 time = 0.05 sec
[ Info: VUMPS 161: obj = +6.478821525323e-02 err = 4.1194192280e-01 time = 0.05 sec
[ Info: VUMPS 162: obj = -2.303254431617e-02 err = 3.8071990310e-01 time = 0.04 sec
[ Info: VUMPS 163: obj = -2.010080764751e-01 err = 3.7801881819e-01 time = 0.04 sec
[ Info: VUMPS 164: obj = -3.545532748718e-01 err = 2.8772339262e-01 time = 0.04 sec
[ Info: VUMPS 165: obj = -3.269217578981e-01 err = 3.3116005000e-01 time = 0.05 sec
[ Info: VUMPS 166: obj = -7.763273380566e-02 err = 4.4233452292e-01 time = 0.05 sec
[ Info: VUMPS 167: obj = -6.297628554394e-02 err = 4.0291320901e-01 time = 0.02 sec
[ Info: VUMPS 168: obj = -6.509753260918e-02 err = 3.7453292998e-01 time = 0.04 sec
[ Info: VUMPS 169: obj = -1.479501367088e-01 err = 3.7646685662e-01 time = 0.06 sec
[ Info: VUMPS 170: obj = -1.936550011338e-01 err = 3.9968563965e-01 time = 0.05 sec
[ Info: VUMPS 171: obj = -2.407761747630e-01 err = 3.8299383049e-01 time = 0.05 sec
[ Info: VUMPS 172: obj = +2.261073305309e-02 err = 4.0104933806e-01 time = 0.03 sec
[ Info: VUMPS 173: obj = -7.166260900992e-02 err = 3.9840210145e-01 time = 0.04 sec
[ Info: VUMPS 174: obj = -1.839330523396e-01 err = 3.7559526845e-01 time = 0.03 sec
[ Info: VUMPS 175: obj = -3.059547394952e-01 err = 3.3999396097e-01 time = 0.04 sec
[ Info: VUMPS 176: obj = -2.995029313597e-01 err = 3.4151945139e-01 time = 0.06 sec
[ Info: VUMPS 177: obj = -1.628200467586e-01 err = 3.9581213200e-01 time = 0.03 sec
[ Info: VUMPS 178: obj = -2.771745987181e-02 err = 3.7799012327e-01 time = 0.04 sec
[ Info: VUMPS 179: obj = -2.450722764418e-01 err = 3.6283636173e-01 time = 0.03 sec
[ Info: VUMPS 180: obj = +2.249435492935e-02 err = 3.8247790534e-01 time = 0.07 sec
[ Info: VUMPS 181: obj = -2.354825221975e-02 err = 4.1422828465e-01 time = 0.03 sec
[ Info: VUMPS 182: obj = -1.142934591710e-01 err = 3.8986670495e-01 time = 0.04 sec
[ Info: VUMPS 183: obj = -1.567308697369e-01 err = 3.9700243050e-01 time = 0.03 sec
[ Info: VUMPS 184: obj = -1.842499532379e-01 err = 3.6532838941e-01 time = 0.05 sec
[ Info: VUMPS 185: obj = -2.514877837893e-01 err = 3.6637072380e-01 time = 0.03 sec
[ Info: VUMPS 186: obj = -5.058982255378e-02 err = 4.0946736692e-01 time = 0.06 sec
[ Info: VUMPS 187: obj = -3.684615096066e-03 err = 4.0659110148e-01 time = 0.03 sec
[ Info: VUMPS 188: obj = -1.225767651180e-01 err = 4.2689031391e-01 time = 0.05 sec
[ Info: VUMPS 189: obj = -2.461679320853e-01 err = 3.6120729961e-01 time = 0.04 sec
[ Info: VUMPS 190: obj = +6.763190603494e-02 err = 3.8401937650e-01 time = 0.03 sec
[ Info: VUMPS 191: obj = +6.358079111108e-02 err = 3.8794532554e-01 time = 0.04 sec
[ Info: VUMPS 192: obj = -2.591627687434e-01 err = 3.4498062399e-01 time = 0.03 sec
[ Info: VUMPS 193: obj = -3.469647117223e-01 err = 3.1920950928e-01 time = 0.07 sec
[ Info: VUMPS 194: obj = -2.328255401437e-01 err = 3.7087112305e-01 time = 0.05 sec
[ Info: VUMPS 195: obj = -1.059181306961e-01 err = 3.9644832676e-01 time = 0.06 sec
[ Info: VUMPS 196: obj = +3.165492219560e-02 err = 3.4664317312e-01 time = 0.04 sec
[ Info: VUMPS 197: obj = -1.060919702177e-01 err = 4.0404073514e-01 time = 0.06 sec
[ Info: VUMPS 198: obj = -7.551485865858e-02 err = 4.1482067043e-01 time = 0.03 sec
[ Info: VUMPS 199: obj = -8.128843893232e-02 err = 3.8193109836e-01 time = 0.07 sec
┌ Warning: VUMPS cancel 200: obj = -1.297030723726e-01 err = 3.8203051738e-01 time = 9.08 sec
└ @ MPSKit ~/Projects/MPSKit.jl/src/algorithms/groundstate/vumps.jl:71
As you can see, VUMPS struggles to converge. On it's own, that is already quite curious. Maybe we can do better using another algorithm, such as gradient descent.
groundstate, cache, delta = find_groundstate(state, H, GradientGrassmann(; maxiter=20));[ Info: CG: initializing with f = 0.249993227089, ‖∇f‖ = 5.1973e-03
[ Info: CG: iter 1: f = -0.104179418829, ‖∇f‖ = 3.1103e-01, α = 1.34e+04, β = 0.00e+00, nfg = 5
[ Info: CG: iter 2: f = -0.168230521916, ‖∇f‖ = 3.1218e-01, α = 1.73e+00, β = 4.36e+01, nfg = 25
[ Info: CG: iter 3: f = -0.279139021454, ‖∇f‖ = 2.5424e-01, α = 1.12e+00, β = 6.09e-01, nfg = 3
[ Info: CG: iter 4: f = -0.351436956991, ‖∇f‖ = 1.8762e-01, α = 8.32e-01, β = 2.88e-01, nfg = 3
[ Info: CG: iter 5: f = -0.394810706473, ‖∇f‖ = 1.5263e-01, α = 7.39e-01, β = 2.23e-01, nfg = 2
[ Info: CG: iter 6: f = -0.421934998173, ‖∇f‖ = 1.1117e-01, α = 6.51e-01, β = 3.64e-01, nfg = 2
[ Info: CG: iter 7: f = -0.432301255729, ‖∇f‖ = 6.0604e-02, α = 5.15e-01, β = 3.63e-01, nfg = 2
[ Info: CG: iter 8: f = -0.436376560912, ‖∇f‖ = 4.3154e-02, α = 3.66e-01, β = 3.53e-01, nfg = 2
[ Info: CG: iter 9: f = -0.438323587926, ‖∇f‖ = 3.4866e-02, α = 2.49e-01, β = 5.20e-01, nfg = 2
[ Info: CG: iter 10: f = -0.439871747915, ‖∇f‖ = 2.4460e-02, α = 3.84e-01, β = 4.02e-01, nfg = 2
[ Info: CG: iter 11: f = -0.440833392787, ‖∇f‖ = 1.7872e-02, α = 3.89e-01, β = 3.60e-01, nfg = 2
[ Info: CG: iter 12: f = -0.441105694369, ‖∇f‖ = 1.6493e-02, α = 1.27e-01, β = 6.83e-01, nfg = 2
[ Info: CG: iter 13: f = -0.441491345325, ‖∇f‖ = 1.4925e-02, α = 1.90e-01, β = 8.48e-01, nfg = 2
[ Info: CG: iter 14: f = -0.441792424830, ‖∇f‖ = 1.2359e-02, α = 2.98e-01, β = 3.23e-01, nfg = 2
[ Info: CG: iter 15: f = -0.442153471150, ‖∇f‖ = 1.0004e-02, α = 4.72e-01, β = 3.45e-01, nfg = 2
[ Info: CG: iter 16: f = -0.442265625493, ‖∇f‖ = 9.2135e-03, α = 1.70e-01, β = 6.20e-01, nfg = 2
[ Info: CG: iter 17: f = -0.442409712129, ‖∇f‖ = 7.1718e-03, α = 3.16e-01, β = 5.69e-01, nfg = 2
[ Info: CG: iter 18: f = -0.442533970670, ‖∇f‖ = 5.9509e-03, α = 3.59e-01, β = 4.16e-01, nfg = 2
[ Info: CG: iter 19: f = -0.442582601670, ‖∇f‖ = 5.6643e-03, α = 1.19e-01, β = 8.47e-01, nfg = 2
┌ Warning: CG: not converged to requested tol: f = -0.442655552067, ‖∇f‖ = 6.1012e-03
└ @ OptimKit ~/.julia/packages/OptimKit/xpmbV/src/cg.jl:103
Convergence is quite slow and even fails after sufficiently many iterations. To understand why, we can look at the transfer matrix spectrum.
transferplot(groundstate, groundstate)We can clearly see multiple eigenvalues close to the unit circle. Our state is close to being non-injective, and represents the sum of multiple injective states. This is numerically very problematic, but also indicates that we used an incorrect ansatz to approximate the groundstate. We should retry with a larger unit cell.
Success
Let's initialize a different initial state, this time with a 2-site unit cell:
state = InfiniteMPS(fill(2, 2), fill(20, 2))2-site InfiniteMPS:
│ ⋮
│ C[2]: TensorMap(ℂ^20 ← ℂ^20)
├── AL[2]: TensorMap((ℂ^20 ⊗ ℂ^2) ← ℂ^20)
├── AL[1]: TensorMap((ℂ^20 ⊗ ℂ^2) ← ℂ^20)
│ ⋮
In MPSKit, we require that the periodicity of the hamiltonian equals that of the state it is applied to. This is not a big obstacle, you can simply repeat the original hamiltonian. Alternatively, the hamiltonian can be constructed directly on a two-site unitcell by making use of MPSKitModels.jl's @mpoham.
# H2 = repeat(H, 2); -- copies the one-site version
H2 = heisenberg_XXX(ComplexF64, Trivial, InfiniteChain(2); spin=1 // 2)
groundstate, envs, delta = find_groundstate(state, H2,
VUMPS(; maxiter=100, tol=1e-12));[ Info: VUMPS init: obj = +4.994527818488e-01 err = 4.7959e-02
[ Info: VUMPS 1: obj = -5.224213331967e-01 err = 3.1869497986e-01 time = 0.07 sec
[ Info: VUMPS 2: obj = -8.703292449817e-01 err = 9.7881092516e-02 time = 0.04 sec
[ Info: VUMPS 3: obj = -8.853872271509e-01 err = 1.1633880994e-02 time = 0.05 sec
[ Info: VUMPS 4: obj = -8.859759226381e-01 err = 5.0303292009e-03 time = 0.06 sec
[ Info: VUMPS 5: obj = -8.861272827748e-01 err = 4.0977568854e-03 time = 0.03 sec
[ Info: VUMPS 6: obj = -8.861876892553e-01 err = 2.8143569786e-03 time = 0.06 sec
[ Info: VUMPS 7: obj = -8.862118265423e-01 err = 2.4768815055e-03 time = 0.07 sec
[ Info: VUMPS 8: obj = -8.862240634280e-01 err = 2.1529722171e-03 time = 0.07 sec
[ Info: VUMPS 9: obj = -8.862295952770e-01 err = 2.0143660733e-03 time = 0.08 sec
[ Info: VUMPS 10: obj = -8.862325590423e-01 err = 1.9696799082e-03 time = 0.07 sec
[ Info: VUMPS 11: obj = -8.862339539915e-01 err = 1.9072844137e-03 time = 0.07 sec
[ Info: VUMPS 12: obj = -8.862347209414e-01 err = 1.9329517027e-03 time = 0.04 sec
[ Info: VUMPS 13: obj = -8.862350788779e-01 err = 1.8912383017e-03 time = 0.08 sec
[ Info: VUMPS 14: obj = -8.862353072621e-01 err = 1.9119217581e-03 time = 0.07 sec
[ Info: VUMPS 15: obj = -8.862354080095e-01 err = 1.8651910333e-03 time = 0.07 sec
[ Info: VUMPS 16: obj = -8.862354997154e-01 err = 1.8780338112e-03 time = 0.07 sec
[ Info: VUMPS 17: obj = -8.862355500667e-01 err = 1.8132834599e-03 time = 0.06 sec
[ Info: VUMPS 18: obj = -8.862356079518e-01 err = 1.8176447019e-03 time = 0.07 sec
[ Info: VUMPS 19: obj = -8.862356026529e-01 err = 1.7897346650e-03 time = 0.06 sec
[ Info: VUMPS 20: obj = -8.862356494807e-01 err = 1.7869452100e-03 time = 0.05 sec
[ Info: VUMPS 21: obj = -8.862357156491e-01 err = 1.7171781823e-03 time = 0.07 sec
[ Info: VUMPS 22: obj = -8.862356705107e-01 err = 1.7618627080e-03 time = 0.05 sec
[ Info: VUMPS 23: obj = -8.862356847446e-01 err = 1.7345893329e-03 time = 0.06 sec
[ Info: VUMPS 24: obj = -8.862356597631e-01 err = 1.7698180368e-03 time = 0.06 sec
[ Info: VUMPS 25: obj = -8.862356369663e-01 err = 1.7832515198e-03 time = 0.07 sec
[ Info: VUMPS 26: obj = -8.862355639238e-01 err = 1.8610433054e-03 time = 0.07 sec
[ Info: VUMPS 27: obj = -8.862355966939e-01 err = 1.8414491439e-03 time = 0.07 sec
[ Info: VUMPS 28: obj = -8.862354448304e-01 err = 1.9601069224e-03 time = 0.07 sec
[ Info: VUMPS 29: obj = -8.862353997121e-01 err = 2.0323064685e-03 time = 0.04 sec
[ Info: VUMPS 30: obj = -8.862352867417e-01 err = 2.0852298701e-03 time = 0.06 sec
[ Info: VUMPS 31: obj = -8.862350688513e-01 err = 2.3296901858e-03 time = 0.08 sec
[ Info: VUMPS 32: obj = -8.862348370300e-01 err = 2.4308916811e-03 time = 0.08 sec
[ Info: VUMPS 33: obj = -8.862348650705e-01 err = 2.5264912559e-03 time = 0.07 sec
[ Info: VUMPS 34: obj = -8.862341340353e-01 err = 2.8806808793e-03 time = 0.07 sec
[ Info: VUMPS 35: obj = -8.862340510957e-01 err = 3.1043746978e-03 time = 0.07 sec
[ Info: VUMPS 36: obj = -8.862334585911e-01 err = 3.2535241129e-03 time = 0.07 sec
[ Info: VUMPS 37: obj = -8.862322313415e-01 err = 4.0963820560e-03 time = 0.05 sec
[ Info: VUMPS 38: obj = -8.862328105793e-01 err = 3.5663760522e-03 time = 0.05 sec
[ Info: VUMPS 39: obj = -8.862317650974e-01 err = 4.3666283036e-03 time = 0.08 sec
[ Info: VUMPS 40: obj = -8.862334734959e-01 err = 3.2172539857e-03 time = 0.06 sec
[ Info: VUMPS 41: obj = -8.862328031496e-01 err = 4.0602562223e-03 time = 0.07 sec
[ Info: VUMPS 42: obj = -8.862348928428e-01 err = 2.8096230226e-03 time = 0.06 sec
[ Info: VUMPS 43: obj = -8.862354347226e-01 err = 2.9469451252e-03 time = 0.07 sec
[ Info: VUMPS 44: obj = -8.862369582324e-01 err = 2.0065666373e-03 time = 0.03 sec
[ Info: VUMPS 45: obj = -8.862376214330e-01 err = 1.5332753632e-03 time = 0.07 sec
[ Info: VUMPS 46: obj = -8.862381576736e-01 err = 9.5235621400e-04 time = 0.07 sec
[ Info: VUMPS 47: obj = -8.862383675148e-01 err = 5.9064456193e-04 time = 0.06 sec
[ Info: VUMPS 48: obj = -8.862384608233e-01 err = 3.4391997505e-04 time = 0.08 sec
[ Info: VUMPS 49: obj = -8.862384963189e-01 err = 1.9914113331e-04 time = 0.07 sec
[ Info: VUMPS 50: obj = -8.862385106659e-01 err = 1.1648442343e-04 time = 0.06 sec
[ Info: VUMPS 51: obj = -8.862385173677e-01 err = 7.0216183061e-05 time = 0.05 sec
[ Info: VUMPS 52: obj = -8.862385211516e-01 err = 5.4936268184e-05 time = 0.07 sec
[ Info: VUMPS 53: obj = -8.862385239261e-01 err = 5.0775181524e-05 time = 0.05 sec
[ Info: VUMPS 54: obj = -8.862385263536e-01 err = 4.9093824356e-05 time = 0.06 sec
[ Info: VUMPS 55: obj = -8.862385287510e-01 err = 4.7160784987e-05 time = 0.05 sec
[ Info: VUMPS 56: obj = -8.862385312350e-01 err = 4.7356857246e-05 time = 0.06 sec
[ Info: VUMPS 57: obj = -8.862385338978e-01 err = 4.6554878190e-05 time = 0.07 sec
[ Info: VUMPS 58: obj = -8.862385367826e-01 err = 4.7893015679e-05 time = 0.05 sec
[ Info: VUMPS 59: obj = -8.862385399464e-01 err = 4.7866639440e-05 time = 0.08 sec
[ Info: VUMPS 60: obj = -8.862385434317e-01 err = 5.0108148399e-05 time = 0.08 sec
[ Info: VUMPS 61: obj = -8.862385472946e-01 err = 5.0701719642e-05 time = 0.06 sec
[ Info: VUMPS 62: obj = -8.862385515924e-01 err = 5.3747205154e-05 time = 0.06 sec
[ Info: VUMPS 63: obj = -8.862385563928e-01 err = 5.4917836680e-05 time = 0.07 sec
[ Info: VUMPS 64: obj = -8.862385617762e-01 err = 5.8736952243e-05 time = 0.07 sec
[ Info: VUMPS 65: obj = -8.862385678319e-01 err = 6.0517251626e-05 time = 0.04 sec
[ Info: VUMPS 66: obj = -8.862385746724e-01 err = 6.5122279370e-05 time = 0.05 sec
[ Info: VUMPS 67: obj = -8.862385824209e-01 err = 6.7603807255e-05 time = 0.08 sec
[ Info: VUMPS 68: obj = -8.862385912344e-01 err = 7.3037975820e-05 time = 0.06 sec
[ Info: VUMPS 69: obj = -8.862386012855e-01 err = 7.6356824733e-05 time = 0.06 sec
[ Info: VUMPS 70: obj = -8.862386127926e-01 err = 8.2675024543e-05 time = 0.08 sec
[ Info: VUMPS 71: obj = -8.862386259979e-01 err = 8.7080551030e-05 time = 0.08 sec
[ Info: VUMPS 72: obj = -8.862386412013e-01 err = 9.4240413691e-05 time = 0.07 sec
[ Info: VUMPS 73: obj = -8.862386587357e-01 err = 1.0001061678e-04 time = 0.06 sec
[ Info: VUMPS 74: obj = -8.862386790008e-01 err = 1.0787560414e-04 time = 0.05 sec
[ Info: VUMPS 75: obj = -8.862387024327e-01 err = 1.1491628128e-04 time = 0.07 sec
[ Info: VUMPS 76: obj = -8.862387295295e-01 err = 1.2350014063e-04 time = 0.07 sec
[ Info: VUMPS 77: obj = -8.862387608070e-01 err = 1.3154045663e-04 time = 0.08 sec
[ Info: VUMPS 78: obj = -8.862387968010e-01 err = 1.4054217709e-04 time = 0.08 sec
[ Info: VUMPS 79: obj = -8.862388379935e-01 err = 1.4918434598e-04 time = 0.08 sec
[ Info: VUMPS 80: obj = -8.862388847728e-01 err = 1.5756496092e-04 time = 0.06 sec
[ Info: VUMPS 81: obj = -8.862389373185e-01 err = 1.6556206078e-04 time = 0.05 sec
[ Info: VUMPS 82: obj = -8.862389955195e-01 err = 1.7207041444e-04 time = 0.05 sec
[ Info: VUMPS 83: obj = -8.862390588490e-01 err = 1.7709969160e-04 time = 0.06 sec
[ Info: VUMPS 84: obj = -8.862391263089e-01 err = 1.8017689240e-04 time = 0.07 sec
[ Info: VUMPS 85: obj = -8.862391964083e-01 err = 1.8083364727e-04 time = 0.06 sec
[ Info: VUMPS 86: obj = -8.862392672643e-01 err = 1.7896061249e-04 time = 0.06 sec
[ Info: VUMPS 87: obj = -8.862393367843e-01 err = 1.7435281981e-04 time = 0.07 sec
[ Info: VUMPS 88: obj = -8.862394029305e-01 err = 1.6733550307e-04 time = 0.07 sec
[ Info: VUMPS 89: obj = -8.862394639791e-01 err = 1.5822580553e-04 time = 0.08 sec
[ Info: VUMPS 90: obj = -8.862395187139e-01 err = 1.4763647565e-04 time = 0.05 sec
[ Info: VUMPS 91: obj = -8.862395665083e-01 err = 1.3613348343e-04 time = 0.07 sec
[ Info: VUMPS 92: obj = -8.862396072866e-01 err = 1.2433109567e-04 time = 0.06 sec
[ Info: VUMPS 93: obj = -8.862396414074e-01 err = 1.1271049944e-04 time = 0.09 sec
[ Info: VUMPS 94: obj = -8.862396695130e-01 err = 1.0166193044e-04 time = 0.08 sec
[ Info: VUMPS 95: obj = -8.862396923871e-01 err = 9.1428430976e-05 time = 0.09 sec
[ Info: VUMPS 96: obj = -8.862397108436e-01 err = 8.2152207117e-05 time = 0.08 sec
[ Info: VUMPS 97: obj = -8.862397256537e-01 err = 7.3876229899e-05 time = 0.06 sec
[ Info: VUMPS 98: obj = -8.862397375053e-01 err = 6.6584753895e-05 time = 0.07 sec
[ Info: VUMPS 99: obj = -8.862397469874e-01 err = 6.0216703928e-05 time = 0.07 sec
┌ Warning: VUMPS cancel 100: obj = -8.862397545895e-01 err = 5.4690768674e-05 time = 6.50 sec
└ @ MPSKit ~/Projects/MPSKit.jl/src/algorithms/groundstate/vumps.jl:71
We get convergence, but it takes an enormous amount of iterations. The reason behind this becomes more obvious at higher bond dimensions:
groundstate, envs, delta = find_groundstate(state, H2,
IDMRG2(; trscheme=truncdim(50), maxiter=20,
tol=1e-12));
entanglementplot(groundstate)We see that some eigenvalues clearly belong to a group, and are almost degenerate. This implies 2 things:
- there is superfluous information, if those eigenvalues are the same anyway
- poor convergence if we cut off within such a subspace
It are precisely those problems that we can solve by using symmetries.
Symmetries
The XXZ Heisenberg hamiltonian is SU(2) symmetric and we can exploit this to greatly speed up the simulation.
It is cumbersome to construct symmetric hamiltonians, but luckily su(2) symmetric XXZ is already implemented:
H2 = heisenberg_XXX(ComplexF64, SU2Irrep, InfiniteChain(2); spin=1 // 2);Our initial state should also be SU(2) symmetric. It now becomes apparent why we have to use a two-site periodic state. The physical space carries a half-integer charge and the first tensor maps the first virtual_space ⊗ the physical_space to the second virtual_space. Half-integer virtual charges will therefore map only to integer charges, and vice versa. The staggering thus happens on the virtual level.
An alternative constructor for the initial state is
P = Rep[SU₂](1 // 2 => 1)
V1 = Rep[SU₂](1 // 2 => 10, 3 // 2 => 5, 5 // 2 => 2)
V2 = Rep[SU₂](0 => 15, 1 => 10, 2 => 5)
state = InfiniteMPS([P, P], [V1, V2]);┌ Warning: Constructing an MPS from tensors that are not full rank
└ @ MPSKit ~/Projects/MPSKit.jl/src/states/infinitemps.jl:149
Even though the bond dimension is higher than in the example without symmetry, convergence is reached much faster:
println(dim(V1))
println(dim(V2))
groundstate, cache, delta = find_groundstate(state, H2,
VUMPS(; maxiter=400, tol=1e-12));52
70
[ Info: VUMPS init: obj = +8.571475316991e-02 err = 4.0869e-01
[ Info: VUMPS 1: obj = -8.841768943048e-01 err = 3.0962778235e-02 time = 8.39 sec
[ Info: VUMPS 2: obj = -8.858941441948e-01 err = 6.8861549505e-03 time = 0.08 sec
[ Info: VUMPS 3: obj = -8.861591970700e-01 err = 3.1438981011e-03 time = 0.19 sec
[ Info: VUMPS 4: obj = -8.862373586553e-01 err = 1.5694150974e-03 time = 0.08 sec
[ Info: VUMPS 5: obj = -8.862665008570e-01 err = 9.6720118128e-04 time = 0.10 sec
[ Info: VUMPS 6: obj = -8.862780790113e-01 err = 7.6876116694e-04 time = 0.11 sec
[ Info: VUMPS 7: obj = -8.862831659742e-01 err = 6.6877783316e-04 time = 0.19 sec
[ Info: VUMPS 8: obj = -8.862855860323e-01 err = 5.3599376124e-04 time = 0.11 sec
[ Info: VUMPS 9: obj = -8.862867663190e-01 err = 4.0660286466e-04 time = 0.16 sec
[ Info: VUMPS 10: obj = -8.862873424424e-01 err = 3.0566269459e-04 time = 0.32 sec
[ Info: VUMPS 11: obj = -8.862876251526e-01 err = 2.2687597738e-04 time = 0.14 sec
[ Info: VUMPS 12: obj = -8.862877641798e-01 err = 1.6630235595e-04 time = 0.28 sec
[ Info: VUMPS 13: obj = -8.862878324776e-01 err = 1.2065962982e-04 time = 0.17 sec
[ Info: VUMPS 14: obj = -8.862878659705e-01 err = 8.6847532670e-05 time = 0.13 sec
[ Info: VUMPS 15: obj = -8.862878823733e-01 err = 6.2151682727e-05 time = 0.16 sec
[ Info: VUMPS 16: obj = -8.862878903974e-01 err = 4.4290628543e-05 time = 0.18 sec
[ Info: VUMPS 17: obj = -8.862878943223e-01 err = 3.1467696209e-05 time = 0.14 sec
[ Info: VUMPS 18: obj = -8.862878962425e-01 err = 2.2309311401e-05 time = 0.17 sec
[ Info: VUMPS 19: obj = -8.862878971826e-01 err = 1.5794437549e-05 time = 0.17 sec
[ Info: VUMPS 20: obj = -8.862878976433e-01 err = 1.1169983677e-05 time = 0.15 sec
[ Info: VUMPS 21: obj = -8.862878978692e-01 err = 7.8938125044e-06 time = 0.17 sec
[ Info: VUMPS 22: obj = -8.862878979802e-01 err = 5.5754187069e-06 time = 0.14 sec
[ Info: VUMPS 23: obj = -8.862878980347e-01 err = 3.9363165672e-06 time = 0.17 sec
[ Info: VUMPS 24: obj = -8.862878980615e-01 err = 2.7782600761e-06 time = 0.18 sec
[ Info: VUMPS 25: obj = -8.862878980747e-01 err = 1.9605416898e-06 time = 0.14 sec
[ Info: VUMPS 26: obj = -8.862878980813e-01 err = 1.3831566356e-06 time = 0.17 sec
[ Info: VUMPS 27: obj = -8.862878980845e-01 err = 9.7563836730e-07 time = 0.17 sec
[ Info: VUMPS 28: obj = -8.862878980861e-01 err = 6.8808237712e-07 time = 0.14 sec
[ Info: VUMPS 29: obj = -8.862878980868e-01 err = 4.8521326515e-07 time = 0.17 sec
[ Info: VUMPS 30: obj = -8.862878980872e-01 err = 3.4211403050e-07 time = 0.18 sec
[ Info: VUMPS 31: obj = -8.862878980874e-01 err = 2.4118986474e-07 time = 0.14 sec
[ Info: VUMPS 32: obj = -8.862878980875e-01 err = 1.7001487151e-07 time = 0.17 sec
[ Info: VUMPS 33: obj = -8.862878980876e-01 err = 1.1983516999e-07 time = 0.16 sec
[ Info: VUMPS 34: obj = -8.862878980876e-01 err = 8.4458211885e-08 time = 0.14 sec
[ Info: VUMPS 35: obj = -8.862878980876e-01 err = 5.9519779420e-08 time = 0.18 sec
[ Info: VUMPS 36: obj = -8.862878980876e-01 err = 4.1941570268e-08 time = 0.14 sec
[ Info: VUMPS 37: obj = -8.862878980877e-01 err = 2.9552481662e-08 time = 0.17 sec
[ Info: VUMPS 38: obj = -8.862878980877e-01 err = 2.0821461973e-08 time = 0.17 sec
[ Info: VUMPS 39: obj = -8.862878980877e-01 err = 1.4668928777e-08 time = 0.13 sec
[ Info: VUMPS 40: obj = -8.862878980877e-01 err = 1.0333740629e-08 time = 0.16 sec
[ Info: VUMPS 41: obj = -8.862878980877e-01 err = 7.2793133017e-09 time = 0.19 sec
[ Info: VUMPS 42: obj = -8.862878980877e-01 err = 5.1274193990e-09 time = 0.14 sec
[ Info: VUMPS 43: obj = -8.862878980877e-01 err = 3.6115696251e-09 time = 0.16 sec
[ Info: VUMPS 44: obj = -8.862878980877e-01 err = 2.5436748324e-09 time = 0.18 sec
[ Info: VUMPS 45: obj = -8.862878980877e-01 err = 1.7914400554e-09 time = 0.13 sec
[ Info: VUMPS 46: obj = -8.862878980877e-01 err = 1.2616075130e-09 time = 0.17 sec
[ Info: VUMPS 47: obj = -8.862878980877e-01 err = 8.8844294429e-10 time = 0.18 sec
[ Info: VUMPS 48: obj = -8.862878980877e-01 err = 6.2563200343e-10 time = 0.14 sec
[ Info: VUMPS 49: obj = -8.862878980877e-01 err = 4.4055009956e-10 time = 0.17 sec
[ Info: VUMPS 50: obj = -8.862878980877e-01 err = 3.1021022428e-10 time = 0.17 sec
[ Info: VUMPS 51: obj = -8.862878980877e-01 err = 2.1842377352e-10 time = 0.13 sec
[ Info: VUMPS 52: obj = -8.862878980877e-01 err = 1.5378883830e-10 time = 0.17 sec
[ Info: VUMPS 53: obj = -8.862878980877e-01 err = 1.0827914915e-10 time = 0.13 sec
[ Info: VUMPS 54: obj = -8.862878980877e-01 err = 7.6233561627e-11 time = 0.17 sec
[ Info: VUMPS 55: obj = -8.862878980878e-01 err = 5.3670361265e-11 time = 0.17 sec
[ Info: VUMPS 56: obj = -8.862878980878e-01 err = 3.7785296901e-11 time = 0.12 sec
[ Info: VUMPS 57: obj = -8.862878980878e-01 err = 2.6597183661e-11 time = 0.15 sec
[ Info: VUMPS 58: obj = -8.862878980878e-01 err = 1.8724144528e-11 time = 0.15 sec
[ Info: VUMPS 59: obj = -8.862878980878e-01 err = 1.3177752530e-11 time = 0.12 sec
[ Info: VUMPS 60: obj = -8.862878980878e-01 err = 9.2726977215e-12 time = 0.15 sec
[ Info: VUMPS 61: obj = -8.862878980878e-01 err = 6.5267922494e-12 time = 0.10 sec
[ Info: VUMPS 62: obj = -8.862878980878e-01 err = 4.5890864081e-12 time = 0.14 sec
[ Info: VUMPS 63: obj = -8.862878980878e-01 err = 3.2289295438e-12 time = 0.10 sec
[ Info: VUMPS 64: obj = -8.862878980878e-01 err = 2.2706304983e-12 time = 0.13 sec
[ Info: VUMPS 65: obj = -8.862878980878e-01 err = 1.5950561789e-12 time = 0.09 sec
[ Info: VUMPS 66: obj = -8.862878980878e-01 err = 1.1189780150e-12 time = 0.09 sec
[ Info: VUMPS conv 67: obj = -8.862878980878e-01 err = 7.7998640130e-13 time = 18.42 sec
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