# Liouvillian gap in ITensor **URL:** \ **Category:** ITensor Julia Questions\ **Created:** [January 25, 2024, 12:28pm UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445 "2024-01-25T12:28:58Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![sourav](https://avatars.discourse-cdn.com/v4/letter/s/a4c791/32.png) [@sourav](https://itensor.discourse.group/u/sourav)\ **Post date:** [January 25, 2024, 12:28pm UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445/1 "2024-01-25T12:28:58Z")
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](https://itensor.github.io/ITensors.jl/stable/examples/DMRG.html)). 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 ```auto ishermitian=false ``` It would be nice if you can comment on this. Best, Sourav. ---
**Author:** ![mtfishman](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/mtfishman/32/11_2.png) [@mtfishman](https://itensor.discourse.group/u/mtfishman)\ **Post date:** [January 25, 2024, 5:35pm UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445/2 "2024-01-25T17:35:57Z")
That would be the easiest thing to try, you’ll have to try it out and see how it works. ---
**Author:** ![sourav](https://avatars.discourse-cdn.com/v4/letter/s/a4c791/32.png) [@sourav](https://itensor.discourse.group/u/sourav)\ **Post date:** [January 25, 2024, 7:55pm UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445/3 "2024-01-25T19:55:50Z")
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: ```auto 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 ---
**Author:** ![mtfishman](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/mtfishman/32/11_2.png) [@mtfishman](https://itensor.discourse.group/u/mtfishman)\ **Post date:** [January 25, 2024, 8:26pm UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445/4 "2024-01-25T20:26:26Z")
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. > **[Biorthonormal transfer-matrix renormalization-group method for non-Hermitian...](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.83.036702)** > > A biorthonormal transfer-matrix renormalization-group (BTMRG) method for non-Hermitian matrices is presented. This BTMRG produces a dual set of biorthonormal bases to construct the renormalized transfer matrix with only half the dimensions of the... 2. > **[Biorthonormal Matrix-Product-State Analysis for Non-Hermitian Transfer-Matrix...](https://arxiv.org/abs/1105.2596)** > > We give a thorough Biorthonormal Matrix-Product-State (BMPS) analysis of the Transfer-Matrix Renormalization-Group (TMRG) for non-Hermitian matrices in the thermodynamic limit. The BMPS is built on a dual series of reduced biorthonormal bases for the... ---
**Author:** ![sourav](https://avatars.discourse-cdn.com/v4/letter/s/a4c791/32.png) [@sourav](https://itensor.discourse.group/u/sourav)\ **Post date:** [January 25, 2024, 9:18pm UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445/5 "2024-01-25T21:18:27Z")
Ok, I see. Thanks a lot for the references. ---
**Author:** ![mtfishman](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/mtfishman/32/11_2.png) [@mtfishman](https://itensor.discourse.group/u/mtfishman)\ **Post date:** [January 29, 2024, 1:17pm UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445/6 "2024-01-29T13:17:28Z")
Here’s a very recent paper on non-Hermitian DMRG: [[2401.15000] Density-matrix renormalization group algorithm for non-Hermitian systems](https://arxiv.org/abs/2401.15000). ---
**Author:** ![sourav](https://avatars.discourse-cdn.com/v4/letter/s/a4c791/32.png) [@sourav](https://itensor.discourse.group/u/sourav)\ **Post date:** [January 31, 2024, 6:49am UTC](https://itensor.discourse.group/t/liouvillian-gap-in-itensor/1445/7 "2024-01-31T06:49:56Z")
Great 🙂 Thanks a lot