Because tensor indices in ITensor have unique identities, before we can make a Hamiltonian or a wavefunction we need to construct a "site set" which will hold the site indices defining the physical Hilbert space:
N = 100 sites = siteinds("S=1",N)
Here we have chosen to create a Hilbert space of N spin 1 sites. The string "S=1" denotes a special Index tag which hooks into a system that knows "S=1" indices have a dimension of 3 and how to create common physics operators like "Sz" for them.
Next we'll make our Hamiltonian matrix product operator (MPO). A very convenient way to do this is to use the OpSum helper type which lets us input a Hamiltonian (or any sum of local operators) in similar notation to pencil-and-paper notation:
ampo = OpSum() for j=1:N-1 ampo += 0.5,"S+",j,"S-",j+1 ampo += 0.5,"S-",j,"S+",j+1 ampo += "Sz",j,"Sz",j+1 end H = MPO(ampo,sites)
In the last line above we convert the OpSum helper object to an actual MPO.
Before beginning the calculation, we need to specify how many DMRG sweeps to do and what schedule we would like for the parameters controlling the accuracy. These parameters are stored within a sweeps object:
sweeps = Sweeps(5) # number of sweeps is 5 maxdim!(sweeps,10,20,100,100,200) # gradually increase states kept cutoff!(sweeps,1E-10) # desired truncation error
The random starting wavefunction
psi0 must be defined in the same Hilbert space as the Hamiltonian, so we construct it using the same collection of site indices:
psi0 = randomMPS(sites,2)
Here we have made a random MPS of bond dimension 2. We could have used a random product state instead, but choosing a slightly larger bond dimension can help DMRG avoid getting stuck in local minima. We could also set psi to some specific initial state using the function
productMPS, which is actually required if we were conserving QNs.
Finally, we are ready to call DMRG:
energy,psi = dmrg(H,psi0,sweeps)
When the algorithm is done, it returns the ground state energy as the variable
energy and an MPS approximation to the ground state as the variable
Below you can find a complete working code that includes all of these steps:
using ITensors let N = 100 sites = siteinds("S=1",N) ampo = OpSum() for j=1:N-1 ampo += 0.5,"S+",j,"S-",j+1 ampo += 0.5,"S-",j,"S+",j+1 ampo += "Sz",j,"Sz",j+1 end H = MPO(ampo,sites) sweeps = Sweeps(5) # number of sweeps is 5 maxdim!(sweeps,10,20,100,100,200) # gradually increase states kept cutoff!(sweeps,1E-10) # desired truncation error psi0 = randomMPS(sites,2) energy,psi = dmrg(H,psi0,sweeps) return end
The following fully-working example shows how to set up a calculation mixing S=1/2 and S=1 spins on every other site of a 1D system. The Hamiltonian involves Heisenberg spin interactions with adjustable couplings between sites of the same spin or different spin.
Note that the only difference from a regular ITensor DMRG calculation is that the
sites array has Index objects which alternate in dimension and in which physical tag type they carry, whether
"S=1". (Try printing out the sites array to see!) These tags tell the OpSum system which local operators to use for these sites when building the Hamiltonian MPO.
using ITensors let N = 100 # Make an array of N Index objects with alternating # "S=1/2" and "S=1" tags on odd versus even sites # (The first argument n->isodd(n) ... is an # on-the-fly function mapping integers to strings) sites = siteinds(n->isodd(n) ? "S=1/2" : "S=1",N) # Couplings between spin-half and # spin-one sites: Jho = 1.0 # half-one coupling Jhh = 0.5 # half-half coupling Joo = 0.5 # one-one coupling ampo = OpSum() for j=1:N-1 ampo += 0.5*Jho,"S+",j,"S-",j+1 ampo += 0.5*Jho,"S-",j,"S+",j+1 ampo += Jho,"Sz",j,"Sz",j+1 end for j=1:2:N-2 ampo += 0.5*Jhh,"S+",j,"S-",j+2 ampo += 0.5*Jhh,"S-",j,"S+",j+2 ampo += Jhh,"Sz",j,"Sz",j+2 end for j=2:2:N-2 ampo += 0.5*Joo,"S+",j,"S-",j+2 ampo += 0.5*Joo,"S-",j,"S+",j+2 ampo += Joo,"Sz",j,"Sz",j+2 end H = MPO(ampo,sites) sweeps = Sweeps(10) maxdim!(sweeps,10,10,20,40,80,100,140,180,200) cutoff!(sweeps,1E-8) psi0 = randomMPS(sites,4) energy,psi = dmrg(H,psi0,sweeps) return end
You can use the OpSum system to make 2D Hamiltonians much in the same way you make 1D Hamiltonians: by looping over all of the bonds and adding the interactions on these bonds to the OpSum.
To help with the logic of 2D lattices, ITensor pre-defines some helper functions which return an array of bonds. Each bond object has an "s1" field and an "s2" field which are the integers numbering the two sites the bond connects. (You can view the source for these functions at this link.)
The two provided functions currently are
triangular_lattice. It is not hard to write your own similar lattice functions as all they have to do is define an array of
ITensors.LatticeBond structs or even a custom struct type you wish to define. We welcome any user contributions of other lattices that ITensor does not currently offer.
Each lattice function takes an optional named argument "yperiodic" which lets you request that the lattice should have periodic boundary conditions around the y direction, making the geometry a cylinder.
Full example code:
using ITensors let Ny = 6 Nx = 12 N = Nx*Ny sites = siteinds("S=1/2", N; conserve_qns = true) # Obtain an array of LatticeBond structs # which define nearest-neighbor site pairs # on the 2D square lattice (wrapped on a cylinder) lattice = square_lattice(Nx, Ny; yperiodic = false) # Define the Heisenberg spin Hamiltonian on this lattice ampo = OpSum() for b in lattice ampo .+= 0.5, "S+", b.s1, "S-", b.s2 ampo .+= 0.5, "S-", b.s1, "S+", b.s2 ampo .+= "Sz", b.s1, "Sz", b.s2 end H = MPO(ampo,sites) state = [isodd(n) ? "Up" : "Dn" for n=1:N] # Initialize wavefunction to a random MPS # of bond-dimension 10 with same quantum # numbers as `state` psi0 = randomMPS(sites,state,20) sweeps = Sweeps(10) maxdim!(sweeps,20,60,100,100,200,400,800) cutoff!(sweeps,1E-8) @show sweeps energy,psi = dmrg(H,psi0,sweeps) return end
ITensor DMRG accepts additional MPS wavefunctions as a optional, extra argument. These additional 'penalty states' are provided as an array of MPS just after the Hamiltonian, like this:
energy,psi3 = dmrg(H,[psi0,psi1,psi2],psi3_init,sweeps)
Here the penalty states are
[psi0,psi1,psi2]. When these are provided, the DMRG code minimizes the energy of the current MPS while also reducing its overlap (inner product) with the previously provided MPS. If these overlaps become sufficiently small, then the computed MPS is an excited state. So by finding the ground state, then providing it to DMRG as a "penalty state" or previous state one can compute the first excited state. Then providing both of these, one can get the second excited state, etc.
A keyword argument called
weight can also be provided to the
dmrg function when penalizing overlaps to previous states. The
weight parameter is multiplied by the overlap with the previous states, so sets the size of the penalty. It should be chosen at least as large as the (estimated) gap between the ground and first excited states. Otherwise the optimal value of the weight parameter is not so obvious, and it is best to try various weights during initial test calculations.
Note that when the system has conserved quantum numbers, a superior way to find excited states can be to find ground states of quantum number (or symmetry) sectors other than the one containing the absolute ground state. In that context, the penalty method used below is a way to find higher excited states within the same quantum number sector.
Full Example code:
using ITensors let N = 20 sites = siteinds("S=1/2",N) h = 4.0 weight = 10*h # use a large weight # since gap is expected to be large # # Use the OpSum feature to create the # transverse field Ising model # # Factors of 4 and 2 are to rescale # spin operators into Pauli matrices # ampo = OpSum() for j=1:N-1 ampo += -4,"Sz",j,"Sz",j+1 end for j=1:N ampo += -2*h,"Sx",j; end H = MPO(ampo,sites) # # Make sure to do lots of sweeps # when finding excited states # sweeps = Sweeps(30) maxdim!(sweeps,10,10,10,20,20,40,80,100,200,200) cutoff!(sweeps,1E-8) noise!(sweeps,1E-6) # # Compute the ground state psi0 # psi0_init = randomMPS(sites,2) energy0,psi0 = dmrg(H,psi0_init,sweeps) println() # # Compute the first excited state psi1 # (Use ground state psi0 as initial state # and as a 'penalty state') # psi1_init = psi0 energy1,psi1 = dmrg(H,[psi0],psi1_init,sweeps; weight) # Check psi1 is orthogonal to psi0 @show inner(psi1,psi0) # # The expected gap of the transverse field Ising # model is given by Eg = 2*|h-1| # # (The DMRG gap will have finite-size corrections.) # println("DMRG energy gap = ",energy1-energy0); println("Theoretical gap = ",2*abs(h-1)); println() # # Compute the second excited state psi2 # (Use ground state psi0 as initial state # and [psi0,psi1] as 'penalty states') # psi2_init = psi0 energy2,psi2 = dmrg(H,[psi0,psi1],psi2_init,sweeps;weight) @show inner(psi2,psi0) @show inner(psi2,psi1) return end