PXP_dmrgs.jl

using ITensors
using Printf
using Random
using LinearAlgebra
using ITensors.HDF5


let
   N = 20  # system size
   maxD = 200 # maximum bond imension
   minD = 20 # minimum bond imension to start with
   initial_energy = -1.1 # intial setting of target energy
   U = 1000 #Redberg interaction
   nk=100 #maximum optimization step

   varray = zeros(1,nk)

   var_thre = 0.1 #threshold for updating ξ
   stop_value = 1e-10 #threshold for stopping

   sites = siteinds("S=1/2",N,conserve_qns=false)


   #define PXP hamiltonian H
   ampo = AutoMPO()

   for j=1:N
      if j==1
        ampo += 1.0,"ProjDn",N,"X",1,"ProjDn",2
      elseif j==N
        ampo += 1.0,"ProjDn",N-1,"X",N,"ProjDn",1
      else
        ampo += 1.0,"ProjDn",j-1,"X",j,"ProjDn",j+1
      end
   end

   H = MPO(ampo, sites)



   #define Redberg interaction H1
   ampo1 = AutoMPO()

   for j=1:N-1
        ampo1 += U,"ProjUp",j,"ProjUp",j+1
   end
   ampo1 += U,"ProjUp",N,"ProjUp",1

   H1 = MPO(ampo1, sites)


   # define Identity MPO H0
   ampo0 = AutoMPO()
   ampo0 += 1.0,"Id",1
   H0 = MPO(ampo0, sites)

   # initial setting of psi 
   numlist = 1:N
   states = [isodd(numlist[n]) ? "Up" : "Dn" for n=1:N]  #Z2 state
   psi0 = productMPS(sites,states) 
   #psi0 = randomMPS(sites,states,2) # random intialization

   # define shift MPO H2
   H2 = H  - (initial_energy)*H0

   @printf("------------------------------------\n")
   psi = copy(psi0)
   for k=1:nk
       @show k
       if k<5          #number adjustable
          maxDim = minD  
       else
          maxDim = maxD
       end
       sweeps = Sweeps(1) #number adjustable
       maxdim!(sweeps, maxDim,maxDim)
       cutoff!(sweeps, 1E-16)
       @show sweeps


       # optimization step
       psi = dmrgs(H2,H0,H,H1,psi0, sweeps)
       psi0 = copy(psi)

       #projecting back to restricted subspace
       orthogonalize!(psi0,1)
       for j=1:N-1
             phi = psi0[j] * psi0[j+1]
             phi0 = copy(phi)
             phi1 = mapprime(phi*op("ProjUp",sites[j]),1 => 0)
             phi2 = mapprime(phi1*op("ProjUp",sites[j+1]),1 => 0)
             phi3= phi0 - phi2

             spec = replacebond!(psi0, j, phi3; maxdim = maxD,
                                              mindim = 1,
                                              cutoff = 1e-15,
                                              eigen_perturbation = nothing,
                                              ortho = "left",
                                              normalize = true,
                                              which_decomp = nothing,
                                              svd_alg = "recursive")

       end


       ampo1 = AutoMPO()
       ampo1 += -1.0,"ProjUp",N,"ProjUp",1
       ampo1 += 1.0,"Id",1
       H_tmp = MPO(ampo1, sites)

       temp = contract(H_tmp,psi0,maxdim = maxDim, normalize=true)
       setprime!(temp,0;plev=1)

       psi0 = copy(temp)
       orthogonalize!(psi0,1)
       normalize!(psi0[1])

       psi =copy(psi0)

       # evaluate energy and variance
       Elist = inner(psi',H,psi)
       Elist1 = inner(psi',H1,psi)
       varlist = inner(H,psi,H,psi) - Elist^2
       @printf("PXP E variance=%.12f,    E=%.12f\n",varlist ,Elist)
       @printf("NNint E=%.12f\n" ,Elist1)


       varray[k]=varlist

       if (varlist<=var_thre)
                H2 = H - (Elist-(-1)^k*3*varlist)*H0 #staggered update to avoid local minima
	        #H2 = H - Elist*H0 
	        var_thre = varlist
                @printf("updated target energy\n")
       end

       #save optimized psi
#       f = h5open(string("data/N",string(N),"_",string(k),".h5"),"w")
#       write(f,"psi",psi)
#       close(f)
       
       if varlist<stop_value
           break
       end

       @printf("------------------------\n")
       for kk=1:k
              @printf("%.12f\n",varray[kk])
       end


   end

   Elist = inner(psi',H,psi)
   Elist1 = inner(psi',H1,psi)
   varlist = inner(H,psi,H,psi) - Elist^2


   @printf("------------------------\n")
   @printf("Final PXP E variance=%.12f,    E=%.12f\n",varlist ,Elist)
   @printf("Final NNint=%.12f\n" ,Elist1)



 end