Liouvillian gap in ITensor

ITensor Julia Questions
1

Dear ITensors team,

I am working on a problem where I need to find our the Liouvillean gap. I have previously worked with ITensor for finding energy gap in Hamiltonian system using ITensor’s DMRG function (following the example given in DMRG Examples · ITensors.jl). My question is: Because Liouvillean is a non-Hermitian object, is it enough to use the same code/setup with the modification that now I put the argument

ishermitian=false

It would be nice if you can comment on this.

Best,
Sourav.

2

That would be the easiest thing to try, you’ll have to try it out and see how it works.

3

Ok Continuing on the same question:

Before working with a Liouvillean, I was just trying to see how good the results I might expect for ground state energy for a non-Hermitian system using ITensor DMRG. To this end, I consider an Ising model with imaginary transverse field. I use the code:

L, hx = 12, 0.5
sites = siteinds("S=1/2", L)
H_b = OpSum()

for j=1:L-1
    H_b += -1.0 ,"Sz", j, "Sz", j+1
    H_b += -1.0*1im*hx, "Sx", j
end
H_b += -1.0*1im*hx,"Sx", L
H = MPO(H_b,sites)
nsweeps = 80
sweeps = Sweeps(nsweeps)
maxdim!(sweeps,10,10,20,20,50,50,100,100,150,150,200,200,400,800,1000)
cutoff!(sweeps,1e-8,1e-8,1e-10,1e-10,1e-12, 1e-12, 1e-12, 1e-12, 1e-12, 1e-14, 1e-14, 1e-14,1e-14,1e-14, 1e-14)
noise!(sweeps,1e-4,1e-4,1e-4,1e-4,1e-6,1e-6,1e-6,1e-8,1e-10,0.0,0.0,0.0,0.,0.,0.)

Energy_obs = DMRGObserver(;energy_tol=1E-11,minsweeps=20,energy_type=Float64)

psi0_init = randomMPS(sites,2)
energy0,psi0 = dmrg(H,psi0_init,sweeps,ishermitian=false;observer=Energy_obs,)
print(energy0)

With this code, the result I get is i) wrong and ii) more alarmingly, each time I run, I get different values of groud state energy. I even tried with minimum sweeps = 10000 apprehending the convergence. But I get the same issue i.e., different runs (which means different initial random MPS) leads to different results.
So, my questions are :
i) Is there any way to improve this code further for non-Hermitian system ?
ii) Is there some criterion for structure of non-Hermitian system where ITensor dmrg with ishermitian=false is supposed to work better?

Best,
Sourav

4

I don’t have much experience with non-Hermitian DMRG, but my intuition is that for the current approach we are using, which just involves changing the local eigensolver from a Hermitian eigensolver (Lanczos) to a non-Hermitian eigensolver (Arnoldi), the closer the operator is to being Hermitian, the better it will work.

Otherwise, there are more advanced techniques for extending DMRG to non-Hermitian operators, for example biorthogonal MPS methods[1][2]. Basically it extends DMRG to target both the dominant left and right eigenvectors of the non-Hermitian operator simultaneously. Unfortunately we don’t have that implemented in ITensor right now but you are free to try to implement it.

  1. ↩︎

  2. ↩︎

5

Ok, I see. Thanks a lot for the references.

6

Here’s a very recent paper on non-Hermitian DMRG: [2401.15000] Density-matrix renormalization group algorithm for non-Hermitian systems.

7

Great :slight_smile: Thanks a lot