DenseMatrix<D_complex> DenseLinearEigenSystem<_Type>::get_tagged_eigenvectors() const
{
if ( TAGGED_INDICES.size() == 0 )
{
std::string problem;
problem = "In DenseLinearEigenSystem.get_tagged_eigenvectors() : there are\n";
problem += "no eigenvalues that have been tagged. This set is empty.\n";
throw ExceptionRuntime( problem );
}
// order of the problem
std::size_t N = EIGENVALUES_ALPHA.size();
// eigenvector storage : size() eigenvectors each of length N
DenseMatrix<D_complex> evecs( TAGGED_INDICES.size(), N, 0.0 );
std::size_t row = 0;
// loop through the tagged set
for ( iter p = TAGGED_INDICES.begin(); p != TAGGED_INDICES.end(); ++p )
{
// get the index of the relevant eigenvalue from the set
std::size_t j = *p;
// put the eigenvector in the matrix
evecs[ row ] = ALL_EIGENVECTORS[ j ];
// next row/eigenvector
++row;
}
return evecs;
}
/**
* \ingroup eigen
* Eigenvectors
* \param m \f$m\f$
* \return a variable matrix with the
* eigenvectors of \f$m\f$ stored in its columns.
*/
dmatrix eigenvectors(const dmatrix &m)
{
if (m.rowsize() != m.colsize())
{
cerr <<
"error -- non square matrix passed to dmatrix eigenvectors(const dmatrix& m)\n";
ad_exit(1);
}
int rmin = m.rowmin();
int rmax = m.rowmax();
dmatrix evecs(rmin, rmax, rmin, rmax);
dvector evals(rmin, rmax);
eigens(m, evecs, evals);
return evecs;
}
void Arnoldi<SCAL>::Calc (int numval, Array<Complex> & lam, int numev,
Array<shared_ptr<BaseVector>> & hevecs,
const BaseMatrix * pre) const
{
static Timer t("arnoldi");
static Timer t2("arnoldi - orthogonalize");
static Timer t3("arnoldi - compute large vectors");
RegionTimer reg(t);
auto hv = a.CreateVector();
auto hv2 = a.CreateVector();
auto hva = a.CreateVector();
auto hvm = a.CreateVector();
int n = hv.FV<SCAL>().Size();
int m = min2 (numval, n);
Matrix<SCAL> matH(m);
Array<shared_ptr<BaseVector>> abv(m);
for (int i = 0; i < m; i++)
abv[i] = a.CreateVector();
auto mat_shift = a.CreateMatrix();
mat_shift->AsVector() = a.AsVector() - shift*b.AsVector();
shared_ptr<BaseMatrix> inv;
if (!pre)
inv = mat_shift->InverseMatrix (freedofs);
else
{
auto itso = make_shared<GMRESSolver<double>> (*mat_shift, *pre);
itso->SetPrintRates(1);
itso->SetMaxSteps(2000);
inv = itso;
}
hv.SetRandom();
hv.SetParallelStatus (CUMULATED);
FlatVector<SCAL> fv = hv.FV<SCAL>();
if (freedofs)
for (int i = 0; i < hv.Size(); i++)
if (! (*freedofs)[i] ) fv(i) = 0;
t2.Start();
// matV = SCAL(0.0); why ?
matH = SCAL(0.0);
*hv2 = *hv;
SCAL len = sqrt (S_InnerProduct<SCAL> (*hv, *hv2)); // parallel
*hv /= len;
for (int i = 0; i < m; i++)
{
cout << IM(1) << "\ri = " << i << "/" << m << flush;
/*
for (int j = 0; j < n; j++)
matV(i,j) = hv.FV<SCAL>()(j);
*/
*abv[i] = *hv;
*hva = b * *hv;
*hvm = *inv * *hva;
for (int j = 0; j <= i; j++)
{
/*
SCAL sum = 0.0;
for (int k = 0; k < n; k++)
sum += hvm.FV<SCAL>()(k) * matV(j,k);
matH(j,i) = sum;
for (int k = 0; k < n; k++)
hvm.FV<SCAL>()(k) -= sum * matV(j,k);
*/
/*
SCAL sum = 0.0;
FlatVector<SCAL> abvj = abv[j] -> FV<SCAL>();
FlatVector<SCAL> fv_hvm = hvm.FV<SCAL>();
for (int k = 0; k < n; k++)
sum += fv_hvm(k) * abvj(k);
matH(j,i) = sum;
for (int k = 0; k < n; k++)
fv_hvm(k) -= sum * abvj(k);
*/
matH(j,i) = S_InnerProduct<SCAL> (*hvm, *abv[j]);
*hvm -= matH(j,i) * *abv[j];
}
*hv = *hvm;
*hv2 = *hv;
SCAL len = sqrt (S_InnerProduct<SCAL> (*hv, *hv2));
if (i<m-1) matH(i+1,i) = len;
*hv /= len;
}
t2.Stop();
t2.AddFlops (double(n)*m*m);
cout << "n = " << n << ", m = " << m << " n*m*m = " << n*m*m << endl;
cout << IM(1) << "\ri = " << m << "/" << m << endl;
Vector<Complex> lami(m);
Matrix<Complex> evecs(m);
Matrix<Complex> matHt(m);
matHt = Trans (matH);
evecs = Complex (0.0);
lami = Complex (0.0);
cout << "Solve Hessenberg evp with Lapack ... " << flush;
LapackHessenbergEP (matH.Height(), &matHt(0,0), &lami(0), &evecs(0,0));
cout << "done" << endl;
for (int i = 0; i < m; i++)
lami(i) = 1.0 / lami(i) + shift;
lam.SetSize (m);
for (int i = 0; i < m; i++)
lam[i] = lami(i);
t3.Start();
if (numev>0)
{
int nout = min2 (numev, m);
hevecs.SetSize(nout);
for (int i = 0; i< nout; i++)
{
hevecs[i] = a.CreateVector();
*hevecs[i] = 0;
for (int j = 0; j < m; j++)
*hevecs[i] += evecs(i,j) * *abv[j];
// hevecs[i]->FVComplex() = Trans(matV)*evecs.Row(i);
}
}
t3.Stop();
}
int main(int argc, char *argv[]) {
if(argc != 6 && argc != 7) {
printf("usage: rrg N n s D seed (id_string)\n");
return 1;
}
time_t t1,t2,tI,tF;
ITensor U,Dg,P,S;
Index ei;
// RRG structure parameters
const int N = atoi(argv[1]); // should be n*(power of 2)
const int n = atoi(argv[2]); // initial blocking size
int w = n; // block size (scales with m)
int ll = 0; // lambda block index
int m = 0; // RG scale factor
// AGSP and subspace parameters
const double t = 0.3; // Trotter temperature
const int M = 100; // num Trotter steps
const int k = 1; // power of Trotter op (just use 1)
const int s = atoi(argv[3]); // formal s param
const int D = atoi(argv[4]); // formal D param
// computational settings
const bool doI = true; // diag restricted Hamiltonian iteratively?
// setup random sampling
std::random_device r;
const int seed = atoi(argv[5]);
fprintf(stderr,"seed is %d\n",seed);
std::mt19937 gen(seed);
std::uniform_real_distribution<double> udist(0.0,1.0);
FILE *sxfl,*syfl,*szfl,*gsfl;
char id[128],sxnm[256],synm[256],sznm[256],gsnm[256];
if(argc == 6) sprintf(id,"rrg-L%d-s%d-D%d",N,s,D);
else sprintf(id,"%s",argv[6]);
strcat(sxnm,id); strcat(sxnm,"-sx.dat");
strcat(synm,id); strcat(synm,"-sy.dat");
strcat(sznm,id); strcat(sznm,"-sz.dat");
strcat(gsnm,id); strcat(gsnm,"-gs.dat");
sxfl = fopen(sxnm,"a");
syfl = fopen(synm,"a");
szfl = fopen(sznm,"a");
gsfl = fopen(gsnm,"a");
// initialize Hilbert subspaces for each level m = 0,...,log(N/n)
vector<SpinHalf> hsps;
for(int x = n ; x <= N ; x *= 2) hsps.push_back(SpinHalf(x));
SpinHalf hs = hsps.back();
// generate product basis over m=0 Hilbert space
auto p = int(pow(2,n));
vector<MPS> V1;
for(int i = 0 ; i < p ; ++i) {
InitState istate(hsps[0],"Dn");
for(int j = 1 ; j <= n ; ++j)
if(i/(int)pow(2,j-1)%2) istate.set(j,"Up");
V1.push_back(MPS(istate));
}
MPS bSpaceL(hsps[0]);
MPS bSpaceR(hsps[0]);
makeVS(V1,bSpaceL,LEFT);
makeVS(V1,bSpaceR,RIGHT);
// Hamiltonian parameters
const double Gamma = 2.0;
vector<double> J(2*(N-1));
fprintf(stdout,"# Hamiltonian terms Jx1,Jy1,Jx2,... (seed=%d)\n",seed);
for(int i = 0 ; i < N-1 ; ++i) {
J[2*i+0] = pow(udist(gen),Gamma);
J[2*i+1] = pow(udist(gen),Gamma);
fprintf(stdout,"%16.14f,%16.14f",J[2*i],J[2*i+1]);
if(i != N-2) fprintf(stdout,",");
}
fprintf(stdout,"\n");
fflush(stdout);
// initialize H for full system and extract block Hamiltonians
AutoMPO autoH(hs);
std::stringstream sts;
auto out = std::cout.rdbuf(sts.rdbuf());
vector<vector<MPO> > Hs(hsps.size());
for(int i = 1 ; i < N ; ++i) {
autoH += (J[2*(i-1)]-J[2*(i-1)+1]),"S+",i,"S+",i+1;
autoH += (J[2*(i-1)]-J[2*(i-1)+1]),"S-",i,"S-",i+1;
autoH += (J[2*(i-1)]+J[2*(i-1)+1]),"S+",i,"S-",i+1;
autoH += (J[2*(i-1)]+J[2*(i-1)+1]),"S-",i,"S+",i+1;
}
auto H = toMPO<ITensor>(autoH,{"Exact",true});
std::cout.rdbuf(out);
for(auto i : args(hsps)) extractBlocks(autoH,Hs[i],hsps[i]);
vector<MPO> prodSz,prodSx,projSzUp,projSzDn,projSxUp,projSxDn;
for(auto& it : hsps) {
auto curSz = sysOp(it,"Sz",2.0).toMPO(); prodSz.push_back(curSz);
auto curSx = sysOp(it,"Sx",2.0).toMPO(); prodSx.push_back(curSx);
auto curSzUp = sysOp(it,"Id").toMPO(); curSzUp.plusEq(curSz); curSzUp /= 2.0;
auto curSzDn = sysOp(it,"Id").toMPO(); curSzDn.plusEq(-1.0*curSz); curSzDn /= 2.0;
auto curSxUp = sysOp(it,"Id").toMPO(); curSxUp.plusEq(curSx); curSxUp /= 2.0;
auto curSxDn = sysOp(it,"Id").toMPO(); curSxDn.plusEq(-1.0*curSx); curSxDn /= 2.0;
projSzUp.push_back(curSzUp); projSzDn.push_back(curSzDn);
projSxUp.push_back(curSxUp); projSxDn.push_back(curSxDn);
}
// approximate the thermal operator exp(-H/t)^k using Trotter
// and MPO multiplication; temperature of K is k/t
time(&tI);
MPO eH(hs);
twoLocalTrotter(eH,t,M,autoH);
auto K = eH;
for(int i = 1 ; i < k ; ++i) {
nmultMPO(eH,K,K,{"Cutoff",eps,"Maxm",MAXBD});
K.Aref(1) *= 1.0/norm(K.A(1));
}
// INITIALIZATION: reduce dimension by sampling from initial basis, either
// bSpaceL or bSpaceR depending on how the merge will work
vector<MPS> Spre;
for(ll = 0 ; ll < N/n ; ll++) {
auto xs = ll % 2 ? 1 : n; // location of dangling Select index
auto cur = ll % 2 ? bSpaceR : bSpaceL;
Index si("ext",s,Select);
// return orthonormal basis of evecs
auto eigs = diagHermitian(-overlapT(cur,Hs[0][ll],cur),P,S,{"Maxm",s});
cur.Aref(xs) *= P*delta(commonIndex(P,S),si);
regauge(cur,xs,{"Truncate",false});
Spre.push_back(cur);
}
time(&t2);
fprintf(stderr,"initialization: %.f s\n",difftime(t2,tI));
// ITERATION: proceed through RRG hierarchy, increasing the scale m
vector<MPS> Spost;
for(m = 0 ; (int)Spre.size() > 1 ; ++m,w*=2) {
fprintf(stderr,"Level %d (w = %d)\n",m,w);
auto hs = hsps[m];
auto DD = D;//max(4,D/(int(log2(N/n)-m)));
auto thr = 1e-8;
Spost.clear();
// EXPAND STEP: for each block, expand dimension of subspace with AGSP operators
for(ll = 0 ; ll < N/w ; ++ll) {
MPO A(hs) , Hc = Hs[m][ll];
MPS pre = Spre[ll] , ret(hs);
int xs = ll % 2 ? 1 : w;
// STEP 1: extract filtering operators A from AGSP K
time(&t1);
restrictMPO(K,A,w*ll+1,DD,ll%2);
time(&t2);
fprintf(stderr,"trunc AGSP: %.f s\n",difftime(t2,t1));
// STEP 2: expand subspace using the mapping A:pre->ret
time(&t1);
ret = applyMPO(A,pre,ll%2,{"Cutoff",eps,"Maxm",MAXBD});
time(&t2);
fprintf(stderr,"apply AGSP: %.f s\n",difftime(t2,t1));
// rotate into principal components of subspace, poxsibly reducing dimension
// and stabilizing numerics, then store subspace in eigenbasis of block H
time(&t1);
diagHermitian(overlapT(ret,ret),U,Dg,{"Cutoff",thr});
time(&t2);
ei = Index("ext",int(commonIndex(Dg,U)),Select);
Dg.apply(invsqrt);
ret.Aref(xs) *= dag(U)*Dg*delta(prime(commonIndex(Dg,U)),ei);
fprintf(stderr,"rotate MPS: %.f s\n",difftime(t2,t1));
auto eigs = diagHermitian(-overlapT(ret,Hs[m][ll],ret),P,S);
ret.Aref(xs) *= P*delta(commonIndex(P,S),ei);
ret.Aref(xs) *= 1.0/sqrt(overlapT(ret,ret).real(ei(1),prime(ei)(1)));
regauge(ret,xs,{"Cutoff",eps});
fprintf(stderr,"max m: %d\n",maxM(ret));
Spost.push_back(ret);
}
// MERGE/REDUCE STEP: construct tensor subspace, sample to reduce dimension
Spre.clear();
for(ll = 0 ; ll < N/w ; ll+=2) {
auto spL = Spost[ll]; // L subspace
auto spR = Spost[ll+1]; // R subspace
// STEP 1: find s lowest eigenpairs of restricted H
time(&t1);
auto tpH = tensorProdContract(spL,spR,Hs[m+1][ll/2]);
tensorProdH<ITensor> resH(tpH);
resH.diag(s,doI);
P = resH.eigenvectors();
time(&t2);
fprintf(stderr,"diag restricted H: %.f s\n",difftime(t2,t1));
// STEP 2: tensor viable sets on each side and reduce dimension
MPS ret(hsps[m+1]);
time(&t1);
tensorProduct(spL,spR,ret,P,(ll/2)%2);
time(&t2);
fprintf(stderr,"tensor product (ll=%d): %.f s\n",ll,difftime(t2,t1));
Spre.push_back(ret);
}
}
// EXIT: extract two lowest energy candidate states to determine gap
auto res = Spre[0];
auto fi = Index("ext",s/2,Select);
vector<MPS> resSz = {res,res};
// project to Sz sectors of the eigenspace
diagHermitian(overlapT(res,prodSz[m],res),U,Dg);
resSz[0].Aref(N) *= U*delta(commonIndex(U,Dg),fi);
diagHermitian(overlapT(res,-1.0*prodSz[m],res),U,Dg);
resSz[1].Aref(N) *= U*delta(commonIndex(U,Dg),fi);
vector<MPS> evecs(2);
for(int i : range(2)) {
auto fc = resSz[i];
// diagonalize H within the Sz sectors
auto eigs = diagHermitian(-overlapT(fc,H,fc),P,S);
fc.Aref(N) *= (P*setElt(commonIndex(P,S)(1)));
fc.orthogonalize({"Cutoff",epx,"Maxm",MAXBD});
fc.normalize();
if(i == 0)
fprintf(stderr,"RRG gs energy: %17.14f\n",overlap(fc,H,fc));
evecs[i] = fc;
}
time(&t2);
Real vz,vx;
vz = overlap(evecs[0],prodSz[m],evecs[0]); vx = overlap(evecs[0],prodSx[m],evecs[0]);
fprintf(stderr,"Vz,vx of 0 is: %17.14f,%17.14f\n",vz,vx);
vz = overlap(evecs[1],prodSz[m],evecs[1]); vx = overlap(evecs[1],prodSx[m],evecs[1]);
fprintf(stderr,"Vz,vx of 1 is: %17.14f,%17.14f\n",vz,vx);
int x1_up = (vx > 0.0 ? 1 : 0);
evecs[0] = exactApplyMPO(evecs[0],projSzUp[m],{"Cutoff",epx});
evecs[0] = exactApplyMPO(evecs[0],projSxUp[m],{"Cutoff",epx});
evecs[1] = exactApplyMPO(evecs[1],projSzDn[m],{"Cutoff",epx});
evecs[1] = exactApplyMPO(evecs[1],(x1_up?projSxUp[m]:projSxDn[m]),{"Cutoff",epx});
for(auto& it : evecs) it.normalize();
fprintf(stderr,"gs candidate energy: %17.14f\nRRG BD ",overlap(evecs[0],H,evecs[0]));
for(const auto& it : evecs) fprintf(stderr,"%d ",maxM(it));
fprintf(stderr,"\telapsed: %.f s\n",difftime(t2,tI));
// CLEANUP: use DMRG to improve discovered evecs
vector<Real> evals(2),e_prev(2);
int max_iter = 30 , used_max = 0;
Real flr = 1e-13 , over_conv = 1e-1 , gap = 1.0 , conv = over_conv*gap , max_conv = 1.0;
for(int i = 0 ; i < (int)evecs.size() ; ++i) evals[i] = overlap(evecs[i],H,evecs[i]);
for(int i = 0 ; (i < 2 || conv < max_conv) && i < max_iter ; ++i) {
e_prev = evals;
time(&t1);
evals = dmrgMPO(H,evecs,8,{"Penalty",0.1,"Cutoff",epx});
time(&t2);
gap = evals[1]-evals[0];
max_conv = 0.0;
for(auto& j : range(2))
if(fabs(e_prev[j]-evals[j]) > max_conv) max_conv = e_prev[j]-evals[j];
fprintf(stderr,"DMRG BD ");
for(const auto& it : evecs) fprintf(stderr,"%3d ",maxM(it));
fprintf(stderr,"\tgap: %e\tconv=%9.2e,%9.2e\telapsed: %.f s\n",gap,
e_prev[0]-evals[0],e_prev[1]-evals[1],difftime(t2,t1));
conv = max(over_conv*gap,flr);
if(i == max_iter) used_max = 1;
}
for(int i = 0 ; i < (int)evecs.size() ; ++i) {
vz = overlap(evecs[i],prodSz[m],evecs[i]); vx = overlap(evecs[i],prodSx[m],evecs[i]);
fprintf(stderr,"Vz,vx of %d is: %12.9f,%12.9f\n",i,vz,vx);
}
evecs[0] = exactApplyMPO(evecs[0],projSzUp[m],{"Cutoff",1e-16});
evecs[0] = exactApplyMPO(evecs[0],projSxUp[m],{"Cutoff",1e-16});
evecs[1] = exactApplyMPO(evecs[1],projSzDn[m],{"Cutoff",1e-16});
evecs[1] = exactApplyMPO(evecs[1],(x1_up?projSxUp[m]:projSxDn[m]),{"Cutoff",1e-16});
for(auto& it : evecs) it.normalize();
for(auto i : range(evecs.size())) evals[i] = overlap(evecs[i],H,evecs[i]);
time(&tF);
auto gsR = evecs[0];
auto ee = measEE(gsR,N/2);
gap = evals[1]-evals[0];
fprintf(stderr,"gs: %17.14f gap: %15.9e ee: %10.8f\n",evals[0],gap,ee);
fprintf(gsfl,"# GS data (L=%d s=%d D=%d seed=%d time=%.f)\n",N,s,D,seed,difftime(tF,tI));
if(used_max) fprintf(gsfl,"# WARNING max iterations reached\n");
fprintf(gsfl,"%17.14f\t%15.9e\t%10.8f\n",evals[0],gap,ee);
// Compute two-point correlation functions in ground state via usual MPS method
fprintf(sxfl,"# SxSx corr matrix (L=%d s=%d D=%d seed=%d)\n",N,s,D,seed);
fprintf(syfl,"# SySy corr matrix (L=%d s=%d D=%d seed=%d)\n",N,s,D,seed);
fprintf(szfl,"# SzSz corr matrix (L=%d s=%d D=%d seed=%d)\n",N,s,D,seed);
for(int i = 1 ; i <= N ; ++i) {
gsR.position(i,{"Cutoff",0.0});
auto SxA = hs.op("Sx",i); auto SyA = hs.op("Sy",i); auto SzA = hs.op("Sz",i);
for(int j = 1 ; j <= N ; ++j) {
if(j <= i) {
fprintf(sxfl,"%15.12f\t",0.0);
fprintf(syfl,"%15.12f\t",0.0);
fprintf(szfl,"%15.12f\t",0.0);
} else {
auto SxB = hs.op("Sx",j); auto SyB = hs.op("Sy",j); auto SzB = hs.op("Sz",j);
fprintf(sxfl,"%15.12f\t",measOp(gsR,SxA,i,SxB,j));
fprintf(syfl,"%15.12f\t",measOp(gsR,SyA,i,SyB,j));
fprintf(szfl,"%15.12f\t",measOp(gsR,SzA,i,SzB,j));
}
}
fprintf(sxfl,"\n");
fprintf(syfl,"\n");
fprintf(szfl,"\n");
}
fclose(sxfl);
fclose(syfl);
fclose(szfl);
fclose(gsfl);
return 0;
}
