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SiteTypes included in this package

This package extends the SiteType collection provided by ITensor, by defining types for (vectorized) density matrices associated to built-in and operators acting on them.

For each of these site types, each operator from the "original", non-vectorized ITensor site type is promoted as a pre- or post-multiplication operator. In other words, if there exists an operator "A" for the "Fermion" type, then you will find operators "A⋅" and "⋅A" (the dot here is a \cdot) available for the "vFermion" site type.

Moreover, the gkslcommutator function allows you to easily define the unitary term (the commutator) in the GKSL equation: given a list of operator names and site indices, it returns an OpSum object representing the $x\mapsto -i[A,x]$ map, for example

julia> gkslcommutator("A", 1)
sum(
  0.0 - 1.0im A⋅(1,)
  0.0 + 1.0im ⋅A(1,)
)

julia> gkslcommutator("A", 1, "B", 3)
sum(
  0.0 - 1.0im A⋅(1,) B⋅(3,)
  0.0 + 1.0im ⋅A(1,) ⋅B(3,)
)

Operators are also vectorized, with the aim of computing expectation values: if $\{a_i\}_{i=1}^{d}$ and $\{r_i\}_{i=1}^{d}$ are, respectively, the coordinates of the operator $A$ and the state $\rho$ with respect to the Gell-Mann basis, then the expectation value $\operatorname{tr}(\rho A)$ is given by $\sum_{i=1}^{d}\overline{a_i}r_i$. Below you can find some operators transformed this way: they are actually "states" in the ITensor formalism, even if they represent operators. Their operator names are usually obtained by prefixing a v to the original ITensor name.

"vS=1/2" SiteType

Site indices with the "vS=1/2" site type represent the density matrix of a $S=1/2$ spin, namely its coefficients with respect to the basis of 2x2 Gell-Mann matrices.

Making a single "vS=1/2" site or collection of N "vS=1/2" sites

s = siteind("vS=1/2")
sites = siteinds("vS=1/2", N)

"vS=1/2" states

The available state names for "vS=1/2" sites are:

  • "Up" spin in the $|{\uparrow}\rangle\langle{\uparrow}|$ state
  • "Dn" spin in the $|{\downarrow}\rangle\langle{\downarrow}|$ state

"vS=1/2" vectorized operators

Vectorized operators or associated with "vS=1/2" sites can be made using the state function, for example

Sx4 = state("vSx", s, 4)
N3 = state("vN", sites[3])

Spin operators:

  • "vId" Identity operator $I_2$
  • "vσx" Pauli x matrix $\sigma^x$
  • "vσy" Pauli y matrix $\sigma^y$
  • "vσz" Pauli z matrix $\sigma^z$
  • "vSx" Spin x operator $S^x = \frac{1}{2} \sigma_x$
  • "vSy" Spin y operator $S^y = \frac{1}{2} \sigma_y$
  • "vSz" Spin z operator $S^z = \frac{1}{2} \sigma_z$
  • "vN" Number operator $N = \frac{1}{2} (I_2+\sigma_z)$

"Osc" SiteType

A SiteType representing a harmonic oscillator. It inherits definitions from the "Qudit" type.

Available keyword arguments for customization:

  • dim (default: 2): dimension of the index (number of oscillator levels)

For example:

sites = siteinds("Osc", N; dm=3)

"vOsc" Operators

Operators associated with "Osc" sites:

Single-oscillator operators:

  • "A" (aliases: "a") annihilation operator
  • "Adag" (aliases: "adag", "a†") creation operator
  • "Asum" equal to $a+a^\dagger$
  • "X" equal to $\frac{1}{\sqrt{2}}(a^\dagger+a)$
  • "Y" equal to $\frac{i}{\sqrt{2}}(a^\dagger-a)$
  • "N" (aliases: "n") number operator

"vOsc" SiteType

Available keyword arguments for customization:

  • dim (default: 2): dimension of the index (number of oscillator levels)

For example:

sites = siteinds("vOsc", N; dim=4)

"vOsc" states

States associated with "vOsc" sites.

  • "n", where n is an integer between 0 and dim-1, gives the Fock state $|n\rangle\langle n|$
  • "ThermEq", with additional parameters temperature and frequency, gives the thermal equilibrium (Gibbs) state ``\operatorname{tr}(\exp(-\frac{\omega}{T}

N))\exp(-\frac{\omega}{T} N)``

  • "X⋅ThermEq", with additional parameters temperature and frequency, gives the thermal equilibrium state (as in the previous point) multiplied by $X=\frac{1}{\sqrt{2}}(a^\dagger+a)$ on the left (this is not actually a state, but it is useful when computing the correlation function of the $X$ operator)

Example:

s = siteind("vOsc", N; dim=4)
rho_eq = state("ThermEq", s; temperature=1.0, frequency=5.0)
fock_st = state("3", s)

"vOsc" Operators

Vectorized operators associated with "vOsc" sites.

Single-oscillator operators (see the operator for "Osc" sites for their meaning):

  • "vA"
  • "vAdag"
  • "vN"
  • "vX"
  • "vY"
  • "vId"

"vFermion" SiteType

Site indices with the "vFermion" SiteType represent spinless fermion sites with the states $|0\rangle$, $|1\rangle$, corresponding to zero fermions or one fermion.

"vFermion" states

The available state names for "vFermion" sites are:

  • "Emp" unoccupied fermion site
  • "Occ" occupied fermion site

"vFermion" operators

Vectorized operators associated with "vFermion" sites:

  • "vId" Identity operator
  • "vN" Density operator
  • "vA" (aliases: "va") Fermion annihilation operator
  • "vAdag" (aliases: "vadag", "vA†", "va†") Fermion creation operator

Note that these creation and annihilation operators do not include Jordan-Wigner strings.

"vElectron" SiteType

The states of site indices with the "vElectron" SiteType correspond to $|0\rangle$, $|{\uparrow}\rangle$, $|{\downarrow}\rangle$, $|{\uparrow\downarrow}\rangle$.

"vElectron" states

The available state names for "vElectron" sites are:

  • "Emp" unoccupied electron site
  • "Up" electron site occupied with one up electron
  • "Dn" electron site occupied with one down electron
  • "UpDn" electron site occupied with two electrons (one up, one down)

"vElectron" operators

Vectorized operators associated with "vElectron" sites:

  • "vId" Identity operator
  • "vNtot" Total density operator
  • "vNup" Up density operator
  • "vNdn" Down density operator
  • "vNupNdn" Product of n↑ and n↓