strange chemical potential of extended boson hubbard model
ALPS users and developers, I am trying to calculate the phase diagram of extended boson hubbard using dmrg. The lattice is the finite one-dimension lattice with open boundaries. The chemical potentials are: u-=E(N)-E(N-1), u+=E(N+1)-E(N). E(N) is the ground energy for a total particle number N. However I found that there is something unusual when calculating the chemical potential: E(N+1)-E(N). When V is not zero, the u+ calculated by DMRG, sparsediag and mps_optim is less than the expected value. For example when L=100, N=50, t=0 and V=0.4, the correct value of u+ is 0.8, however using dmrg(mps and sparsediag) the value is 0.4 . I can obtain the correct value using u+=E(N+2)-E(N+1) when choosing L=41 and N=21 instead. However within L=101 and N=101, even choose u+=E(N+2)-E(N+1) the result is still 0.4. It seems that when V does not equal to zero the calculated chemical potential is related to specific total particle numbers. Xiao Liang
On 26 May 2016, at 03:29, 梁霄 <lxxhlb@mail.ustc.edu.cn> wrote:
ALPS users and developers, I am trying to calculate the phase diagram of extended boson hubbard using dmrg. The lattice is the finite one-dimension lattice with open boundaries. The chemical potentials are: u-=E(N)-E(N-1), u+=E(N+1)-E(N). E(N) is the ground energy for a total particle number N. However I found that there is something unusual when calculating the chemical potential: E(N+1)-E(N). When V is not zero, the u+ calculated by DMRG, sparsediag and mps_optim is less than the expected value. For example when L=100, N=50, t=0 and V=0.4, the correct value of u+ is 0.8, however using dmrg(mps and sparsediag) the value is 0.4 .
Why should it be 0.8? If I place a particle on all even sites (0, 2, … 98) and then add one more at site 99 the energy only goes up by 0.4.
I can obtain the correct value using u+=E(N+2)-E(N+1) when choosing L=41 and N=21 instead.
Indeed. Now you have an odd length and you place particles at sites 0, 2, … 40. The next particle has to go in-between to particles on the sites 1, 3, …. 39 and thus you pay 2*V
However within L=101 and N=101, even choose u+=E(N+2)-E(N+1) the result is still 0.4.
Once more, to reduce the total energy you place the extra particle at site 0 or 100.
It seems that when V does not equal to zero the calculated chemical potential is related to specific total particle numbers.
Indeed, it is, but that is due to the length (odd/even effects) and boundary conditions. ALPS gives the correct values Matthias Troyer
Matthias Troyer, Very appreciate your helpful replying. I changed the boundary condition to periodic and achieve the expected answer now. Thanks. I found that when using UNITCELL(simple1d) with periodic boundaries, the correlations cannot be measured. Hence I have to define a custom graph and ALPS works the way as I wished. Xiao Liang
-----Original Messages----- From: "Matthias Troyer" <troyer@phys.ethz.ch> Sent Time: 2016-05-26 23:51:31 (Thursday) To: comp-phys-alps-users@lists.phys.ethz.ch Cc: Subject: Re: [ALPS-users] strange chemical potential of extended boson hubbard model
On 26 May 2016, at 03:29, 梁霄 <lxxhlb@mail.ustc.edu.cn> wrote:
ALPS users and developers, I am trying to calculate the phase diagram of extended boson hubbard using dmrg. The lattice is the finite one-dimension lattice with open boundaries. The chemical potentials are: u-=E(N)-E(N-1), u+=E(N+1)-E(N). E(N) is the ground energy for a total particle number N. However I found that there is something unusual when calculating the chemical potential: E(N+1)-E(N). When V is not zero, the u+ calculated by DMRG, sparsediag and mps_optim is less than the expected value. For example when L=100, N=50, t=0 and V=0.4, the correct value of u+ is 0.8, however using dmrg(mps and sparsediag) the value is 0.4 .
Why should it be 0.8? If I place a particle on all even sites (0, 2, … 98) and then add one more at site 99 the energy only goes up by 0.4.
I can obtain the correct value using u+=E(N+2)-E(N+1) when choosing L=41 and N=21 instead.
Indeed. Now you have an odd length and you place particles at sites 0, 2, … 40. The next particle has to go in-between to particles on the sites 1, 3, …. 39 and thus you pay 2*V
However within L=101 and N=101, even choose u+=E(N+2)-E(N+1) the result is still 0.4.
Once more, to reduce the total energy you place the extra particle at site 0 or 100.
It seems that when V does not equal to zero the calculated chemical potential is related to specific total particle numbers.
Indeed, it is, but that is due to the length (odd/even effects) and boundary conditions. ALPS gives the correct values
Matthias Troyer
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participants (2)
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Matthias Troyer
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梁霄