Truncation errors in calculating ground state energy of the Hubbard Ladder

Question

Hi,

I am playing around the Hubbard ladder model using the code formula of the extHubbard model. The Hubbard ladder model is the most simple one with intra-chain nearest neighbour hopping t , inter-chain hopping t1 and one site interaction U. The code snipet of the model is the following:

auto sites = Hubbard(N);
/*
  *Create the Hamiltonian using AutoMPO
  */
auto ampo = AutoMPO(sites);
for(int i = 1; i <= N; ++i) 
    {
    ampo += U,"Nupdn",i;
    }
/* intra-chain hopping t */
for(int b = 1; b < N-1; ++b)  
    {
    ampo += -t,"Cdagup",b,"Cup",b+2;
    ampo += -t,"Cdagup",b+2,"Cup",b;
    ampo += -t,"Cdagdn",b,"Cdn",b+2;
    ampo += -t,"Cdagdn",b+2,"Cdn",b;
    }
/* inter-chain hopping t1 */
for(int b = 1; b < N; ++b)
    {
    if( (b & 0x1) )
      {
       ampo += -t1,"Cdagup",b,"Cup",b+1;
       ampo += -t1,"Cdagup",b+1,"Cup",b;
       ampo += -t1,"Cdagdn",b,"Cdn",b+1;
       ampo += -t1,"Cdagdn",b+1,"Cdn",b;
      }
    }

auto H = IQMPO(ampo);

And I use the Neel state as the initial wavefunction mps.

The result does converge but with truncation error 10^(-6). Comparing with the 10^(-12) error in the result of extHubbard, this is not good. I have tried to improve the result by increasing sweeps.maxm() to 1000 and the trunc error stays at 10^(-6) until the 20th sweeps.

Any suggestions ? Thanks a lot for your help!

Best wishes

Deping Zhang

Answer 1

Hi Deping,
I tried it out for myself and there doesn't seem to be anything wrong with what you're observing for this system. You didn't mention what U you are using, but for small to moderate U and this high of a filling I think you can encounter systems with very high entanglement.

I tried the parameters t=1.0, t1=0.5, and U=3.0 and found similar results to yours for maxm=1000. But then I increased the maxm up to 2000 and managed to reduce the truncation error below 3E-8.

It's far from a perfect comparison but a very quick literature search revealed a paper from May 2000 by Rommer, White, and Scalapino studying t-J ladders. They kept up to 2400 states and still the best truncation error they observed was only 1E-7.

If you try cranking U up much higher, say 10 times t, then you should find you can get much smaller errors for bond dimensions in the few hundreds (because the entanglement is a lot lower).

The exthubbard results may have been unintentionally misleading; perhaps the combination of being 1d, being only at quarter filling, and having strong interactions both U and V1 conspire to lower the entanglement.

Let me know if you find some literature on this system suggesting the entanglement should be lower -

Miles

Comments

Thank you very much, miles.