An important technique in DMRG calculations of quantum Hamiltonians is the conservation of quantum numbers. Examples of these are the total number of particles of a model of fermions, or the total of all $S^z$ components of a system of spins. Not only can conserving quantum numbers make DMRG calculations run more quickly and use less memory, but it can be important for simulating physical systems with conservation laws and for obtaining ground states in different symmetry sectors. Note that ITensor currently only supports Abelian quantum numbers.
Setting up a quantum-number conserving DMRG calculation in ITensor requires only very small changes to a DMRG code. The main changes are:
- using tensor indices (
Indexobjects) which carry quantum number (QN) information to build your Hamiltonian and initial state
- initializing your MPS to have well-defined total quantum numbers
Importantly, the total QN of your state throughout the calculation will remain the same as the initial state passed to DMRG. The total QN of your state is not set separately, but determined implicitly from the initial QN of the state when it is first constructed.
Of course, your Hamiltonian should conserve all of the QN's that you would like to use. If it doesn't, you will get an error when you try to construct it out of the QN-enabled tensor indices.
Let's see how to make these two changes to the DMRG Tutorial code from the previous section. At the end, we will put together these changes for a complete, working code.
Change 1: QN Site Indices
To make change (1), we will change the line
sites = siteinds("S=1",N)
by setting the
conserve_qns keyword argument to
sites = siteinds("S=1",N; conserve_qns=true)
conserve_qns=true tells the
siteinds function to conserve every possible quantum number associated to the site type (which is
"S=1" in this example). For $S=1$ spins, this will turn on total-$S^z$ conservation. (For other site types that conserve multiple QNs, there are specific keyword arguments available to track just a subset of conservable QNs.) We can check this by printing out some of the site indices, and seeing that the subspaces of each
Index are labeled by QN values:
@show sites @show sites
sites = (dim=3|id=794|"S=1,Site,n=1") <Out> 1: QN("Sz",2) => 1 2: QN("Sz",0) => 1 3: QN("Sz",-2) => 1 sites = (dim=3|id=806|"S=1,Site,n=2") <Out> 1: QN("Sz",2) => 1 2: QN("Sz",0) => 1 3: QN("Sz",-2) => 1
In the sample output above, note that in addition to the dimension of these indices being 3, each of the three settings of the Index have a unique QN associated to them. The number after the QN on each line is the dimension of that subspace, which is 1 for each subspace of the Index objects above. Note also that
"Sz" quantum numbers in ITensor are measured in units of $1/2$, so
QN("Sz",2) corresponds to $S^z=1$ in conventional physics units.
Change 2: Initial State
To make change (2), instead of constructing the initial MPS
psi0 to be an arbitrary, random MPS, we will make it a specific state with a well-defined total $S^z$. So we will replace the line
psi0 = randomMPS(sites,10)
by the lines
state = [isodd(n) ? "Up" : "Dn" for n=1:N] psi0 = MPS(sites,state)
The first line of the new code above makes an array of strings which alternate between
"Dn" on odd and even numbered sites. These names
"Dn" are special values associated to the
"S=1" site type which indicate up and down spin values. The second line takes the array of site Index objects
sites and the array of strings
state and returns an MPS which is a product state (classical, unentangled state) with each site's state given by the strings in the
state array. In this example,
psi0 will be a Neel state with alternating up and down spins, so it will have a total $S^z$ of zero. We could check this by computing the quantum-number flux of
@show flux(psi0) # Output: flux(psi0) = QN("Sz",0)
The above example shows the case of setting a total "Sz" quantum number of zero, since the initial state alternates between "Up" and "Dn" on every site with an even number of sites.
To obtain other total QN values, just set the initial state to be one which has the total QN you want. To be concrete let's take the example of a system with
N=10 sites of $S=1$ spins.
For example if you want a total "Sz" of +20 (=
QN("Sz",20)) in ITensor units, or $S^z=10$ in physical units, for a system with 10 sites, use the initial state:
state = ["Up" for n=1:N] psi0 = MPS(sites,state)
Or to initialize this 10-site system to have a total "Sz" of +16 in ITensor units ($S^z=8$ in physical units):
state = ["Dn","Up","Up","Up","Up","Up","Up","Up","Up","Up"] psi0 = MPS(sites,state)
would work (as would any
state with one "Dn" and nine "Up"'s in any order). Or you could initialize to a total "Sz" of +18 in ITensor units ($S^z=9$ in physical units) as
state = ["Z0","Up","Up","Up","Up","Up","Up","Up","Up","Up"] psi0 = MPS(sites,state)
where "Z0" refers to the $S^z=0$ state of a spin-one spin.
Finally, the same kind of logic as above applies to other physical site types, whether "S=1/2", "Electron", etc.
Let's take the DMRG Tutorial code from the previous section and make the changes discussed above, to turn it into a code which conserves the total $S^z$ quantum number throughout the DMRG calculation. The resulting code is:
using ITensors let N = 100 sites = siteinds("S=1",N;conserve_qns=true) os = OpSum() for j=1:N-1 os += "Sz",j,"Sz",j+1 os += 1/2,"S+",j,"S-",j+1 os += 1/2,"S-",j,"S+",j+1 end H = MPO(os,sites) state = [isodd(n) ? "Up" : "Dn" for n=1:N] psi0 = MPS(sites,state) @show flux(psi0) nsweeps = 5 maxdim = [10,20,100,100,200] cutoff = [1E-10] energy, psi = dmrg(H,psi0; nsweeps, maxdim, cutoff) return end