# DMRG for obtaining multi excited states with conserved sz

Dear all,

I am currently working on applying DMRG to compute the ground-state energy and numerous excited state energies of a finite-size (N=6, 12) spin-1 chain subjected to a magnetic field (Bz), assuming conserved sz. Following this, I aim to determine the partition function to extract the free energy and magnetization at low temperatures. However, when utilizing the DMRG method, I consistently obtain zero magnetization for all values of the magnetic field.

Here’s the function that I’ve been using:

``````# Calculate free energy and magnetization
function calculate_FreeE_and_Mag(N, J, B_min, B_max, uB, sweeps, linkdim, cutoff, noise, T)
sites = siteinds("S=1", N; conserve_sz=true)
sB = Int((B_max - B_min) / uB)  # number of field steps
h = zeros(sB + 1)                       # magnetic field
F_values = zeros(sB + 1)          # free energy
M_values = zeros(sB + 1)         # magnetization

for (m_B, Bz) in enumerate(B_min:uB:B_max)
h[m_B] = Bz

statei = ["Emp" for n in 1:N]  # assuming conserved sz with permutation 3
for i in 1:3:N
statei[i] = "Up"
end
for i in 2:3:N
statei[i] = "Z0"
end
for i in 3:3:N
statei[i] = "Dn"
end

ψi = randomMPS(sites, statei, linkdim)
H = H_XXXchain(N, J, J2z, Bz, sites)

E0, ψ0 = dmrg(H, ψi, sweeps; write_when_maxdim_exceeds=1000)
En = [E0]
ψn = [ψ0]
for i in 1:12  # number of exited states for N=12
E, ψ = dmrg(H, ψn, ψi, sweeps; write_when_maxdim_exceeds=1000)
@show i   E
println()
push!(En,E)
push!(ψn,ψ)
end
# ------------------------------------------------------------------
Z = sum(exp.(-En / T))   # partition function
F_values[m_B] = -T * log(Z)   # free energy
# Calculate magnetization using finite difference
if m_B > 1
M_values[m_B] = (1/N)*(F_values[m_B - 1] - F_values[m_B]) / uB  # magnetization
end

println("Bz = \$Bz, Free Energy = \$(F_values[m_B]), Magnetization = \$(M_values[m_B])")
end

return F_values, M_values, h
end
``````

However, when comparing this to the exact diagonalization (ED) method for N=12, the ED method yields accurate magnetization values for each magnetic field and temperature. I use following expression for ED method:

``````H_tensor = prod(H)
eigen_E, _ = eigen(H_tensor)
Z = sum(exp.(-eigen_E / T))
F_values[m_B] = -T * log(Z)
# Calculate magnetization using finite difference
if m_B > 1
M_values[m_B] = (1/N)*(F_values[m_B - 1] - F_values[m_B]) / uB
end
``````

Unfortunately, in my case, ED does not work for `N>=12`. In fact, I get following error:

``````Contraction resulted in ITensor with 15 indices, which is greater
than or equal to the ITensor order warning threshold 14.
You can modify the threshold with macros like `@set_warn_order N`,
`@reset_warn_order`, and `@disable_warn_order` or functions like
`ITensors.set_warn_order(N::Int)`, `ITensors.reset_warn_order()`, and
`ITensors.disable_warn_order()`.
Stacktrace:
[1] _contract(A::ITensor, B::ITensor)
@ ITensors ~/.julia/packages/ITensors/Gf9aD/src/tensor_operations/tensor_algebra.jl:20
[2] contract(A::ITensor, B::ITensor)
@ ITensors ~/.julia/packages/ITensors/Gf9aD/src/tensor_operations/tensor_algebra.jl:104
[3] *
@ ~/.julia/packages/ITensors/Gf9aD/src/tensor_operations/tensor_algebra.jl:91 [inlined]
[4] mul_prod
@ ./reduce.jl:35 [inlined]
[5] BottomRF
@ ./reduce.jl:86 [inlined]
[6] _foldl_impl
@ ./reduce.jl:62 [inlined]
[7] foldl_impl(op::Base.BottomRF{typeof(Base.add_sum)}, nt::Base._InitialValue, itr::ITensor)
@ Base ./reduce.jl:48 [inlined]
[8] mapfoldl_impl
@ ./reduce.jl:44 [inlined]
[9] mapfoldl
@ ./reduce.jl:175 [inlined]
[10] mapreduce
@ ./reduce.jl:307 [inlined]
[11] prod
@ ./reduce.jl:620 [inlined]
``````

Despite having 64 gigabytes of RAM, assuming conserved sz, and implementing the term `ITensors.disable_warn_order()`, ED for N=12 spin-1 particles unexpectedly halts due to memory limitations.

My guess is that the DMRG should also provide accurate results when sz is conserved. It’s worth noting that when I consider non-conserved sz, the DMRG works and provides accurate magnetization for any magnetic field and temperature. However, when conserved sz is considered, the computation time significantly decreases.

I’ve also tried with the predefined code purification.jl, but unfortunately, I couldn’t obtain accurate magnetization values for non-zero magnetic fields. Despite this, using the thermal partition function, I can still obtain other thermodynamic quantities.

If you have any insights or suggestions on how to address the issues for ED and DMRG and improve the accuracy of the magnetization results in the DMRG method, please let me know.

Thanks