Hello,
I wanted to make sure c,cdag
on Fermion
sites worked in the way I expected, so I made a 3-site MPS, occupied on every site, and did fermion annihilation on sites 1 and 2, in both orders. (Code below.)
I can tell from the indices that apply(c2, ψ)
(where c2
is annihilation on site 2 and ψ
is my MPS) does something nonlocal. I assume that nonlocal something is applying the Jordan-Wigner string—there’re bits and pieces of documentation that hint that direction, and a quick poke at the code does too. (Maybe I’m reading those hints incorrectly!)
But then I do
order1 = apply(c2, apply(c1, ψ))
order2 = apply(c1, apply(c2, ψ))
@show inner( order1, order2)
and find that the inner product is 1, where I’d expect -1! (Again, full code and output below.)
My suspicion is that there’s an off-by-one error, and the J-W string with c_j & c^\dagger_j mistakenly includes site j, where it should only include everything to the right of (but not including) j. Part of why I think this is that when I start with a state
φ = MPS(ss, ["0", "1", "0"])
and do c^\dagger_1 or c^\dagger_3, one or the other should pick up a minus sign, but in fact neither picks up a minus sign—or, rather, in both cases the tensors pick up two minus signs. (I’m fairly confused about which way the JW string is going.) (Again, full code and output below.)
It’s tempting to say this is a bug, but I don’t understand the intended function of the code well enough to say! Maybe my expectations are wrong?
So—what’s the story? Is there something basic I’m missing?
Best,
Christopher
Full code:
using ITensors
using ITensorMPS
using Pkg
versioninfo()
Pkg.status("ITensors")
Pkg.status("ITensorMPS")
ss = siteinds("Fermion", 3)
ψ = MPS(ss, ["1", "1", "1"])
@show ψ
c1 = op("c", ss[1])
c2 = op("c", ss[2])
@show apply(c2, ψ)
order1 = apply(c2, apply(c1, ψ))
order2 = apply(c1, apply(c2, ψ))
@show inner( order1, order2)
println("=====")
φ = MPS(ss, ["0", "1", "0"])
@show φ[1] |> array
@show φ[2] |> array
@show φ[3] |> array
println("-----")
cdag1φ = apply(op("cdag", ss[1]), φ)
@show cdag1φ[1] |> array
@show cdag1φ[2] |> array
@show cdag1φ[3] |> array
println("-----")
cdag3φ = apply(op("cdag", ss[3]), φ)
@show cdag3φ[1] |> array
@show cdag3φ[2] |> array
@show cdag3φ[3] |> array
Output:
Julia Version 1.8.2
Commit 36034abf260 (2022-09-29 15:21 UTC)
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 8 × 11th Gen Intel(R) Core(TM) i5-1135G7 @ 2.40GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-13.0.1 (ORCJIT, tigerlake)
Threads: 1 on 8 virtual cores
Status `~/.julia/environments/v1.8/Project.toml`
[9136182c] ITensors v0.6.16
Status `~/.julia/environments/v1.8/Project.toml`
[0d1a4710] ITensorMPS v0.2.4
ψ = MPS
[1] ((dim=2|id=578|"Fermion,Site,n=1"), (dim=1|id=379|"Link,l=1"))
[2] ((dim=1|id=379|"Link,l=1"), (dim=2|id=238|"Fermion,Site,n=2"), (dim=1|id=102|"Link,l=2"))
[3] ((dim=1|id=102|"Link,l=2"), (dim=2|id=11|"Fermion,Site,n=3"))
apply(c2, ψ) = MPS
[1] ((dim=2|id=578|"Fermion,Site,n=1"), (dim=1|id=283|"Link,l=1"))
[2] ((dim=2|id=238|"Fermion,Site,n=2"), (dim=1|id=283|"Link,l=1"), (dim=1|id=540|"Link,l=2"))
[3] ((dim=2|id=11|"Fermion,Site,n=3"), (dim=1|id=540|"Link,l=2"))
inner(order1, order2) = 1.0
=====
φ[1] |> array = [1.0; 0.0;;]
φ[2] |> array = [0.0 1.0;;;]
φ[3] |> array = [1.0 0.0]
-----
cdag1φ[1] |> array = [0.0; -1.0;;]
cdag1φ[2] |> array = [0.0; -1.0;;;]
cdag1φ[3] |> array = [1.0; 0.0;;]
-----
cdag3φ[1] |> array = [1.0; 0.0;;]
cdag3φ[2] |> array = [0.0; -1.0;;;]
cdag3φ[3] |> array = [0.0; -1.0;;]