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main / formulas / spinless_correlator_mps

Measure a spinless fermion correlator from an MPS wavefunction

Contributed by Jon Spalding, UC Riverside— Oct 12, 2016

For more background on working with fermions in ITensor and Jordan-Wigner string, see the tutorial on fermions.

Sample code:

// Given an MPS called "psi",
// constructed from a Fermion SiteSet "sites"

// Consider a pair of fermionic operators,
// Cdag and C, for spinless fermions.
// If Cdag(i) and C(j) operate at sites separated by some arbitrary
// distance, so that i < j, then in order to represent them
// as bosonic operators Adag(i) and A(j), we need to include
// a chain of "string" (F) operators that preserves the commutation
// relation for Cdag and C.

int N = 20;
auto sites = Fermion(N);
auto t = 0.5;
auto ampo = AutoMPO(sites);
for( auto n : range1(N-1) )
    ampo += -t,"Cdag",n,"C",n+1;
    ampo += -t,"Cdag",n+1,"C",n;
auto H = toMPO(ampo);
auto state = InitState(sites,"Emp");
for( auto n : range1(N) ) if( n%2==0 ) state.set(n,"Occ");
auto sweeps = Sweeps(5); //number of sweeps is 5
sweeps.maxdim() = 10,20,100,100,200;
sweeps.cutoff() = 1E-10;
auto [energy,psi] = dmrg(H,randomMPS(state),sweeps,"Silent");

auto i = 5;
auto j = 10;

auto Adag_i = op(sites,"Adag",i);
auto A_j = op(sites,"A",j);

//'gauge' the MPS to site i
//any 'position' between i and j, inclusive, would work here

auto psidag = dag(psi);

//index linking i to i-1:
auto li_1 = leftLinkIndex(psi,i);
auto Cij = prime(psi(i),li_1)*Adag_i*psidag(i);
for(int k = i+1; k < j; ++k)
    Cij *= psi(k);
    Cij *= op(sites,"F",k); //Jordan-Wigner string
    Cij *= psidag(k);
//index linking j to j+1:
auto lj = rightLinkIndex(psi,j);
Cij *= prime(psi(j),lj);
Cij *= A_j;
Cij *= psidag(j);

auto result = elt(Cij); //or eltC(Cij) if expecting complex


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