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SiteTypes Included with ITensor

"S=1/2" SiteType

Site indices with the "S=1/2" site type represent $S=1/2$ spins with the states $|\!\uparrow\rangle$, $|\!\downarrow\rangle$.

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

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

Available keyword arguments for enabling and customizing quantum numbers (QN) subspaces:

  • conserve_qns (default: false): conserve total $S^z$
  • conserve_sz (default: conserve_qns): conserve total $S^z$
  • conserve_szparity (default: false): conserve total $S^z$ modulo two
  • qnname_sz (default: "Sz"): name of total $S^z$ QN
  • qnname_szparity (default: "SzParity"): name of total $S^z$ modulo two QN

For example:

sites = siteinds("S=1/2",N; conserve_szparity=true, qnname_szparity="SzP")

Operators associated with "S=1/2" sites can be made using the op function, for example

Sz = op("Sz",s)
Sz4 = op("Sz",sites[4])

Available operators are exactly the same as those for the "Qubit" site type. Please see the list of "Qubit" operators below.

"Qubit" SiteType

Site indices with the "Qubit" site type represent qubits with the states $|0\rangle$, $|1\rangle$.

Making a single "Qubit" site or collection of N "Qubit" sites

s = siteind("Qubit")
sites = siteinds("Qubit",N)

Available keyword arguments for enabling and customizing quantum numbers (QN) subspaces:

  • conserve_qns (default: false): conserve total qubit parity
  • conserve_parity (default: conserve_qns): conserve total qubit parity
  • conserve_number (default: false): conserve total qubit number
  • qnname_parity (default: "Parity"): name of total qubit parity QN
  • qnname_number (default: "Number"): name of total qubit number QN

For example:

sites = siteinds("Qubit",N; conserve_parity=true)

"Qubit" and "S=1/2" States

The available state names for "Qubit" sites are:

  • "0" (aliases: "Z+", "Zp", "Up", "↑") Qubit in the 0 state
  • "1" (aliases: "Z-", "Zm", "Dn", "↓") Qubit in the 1 state
  • "+" (aliases: "X+", "Xp") Qubit in the $|+\rangle$ state (+1 eigenvector of $\sigma_x$)
  • "+" (aliases: "X-", "Xm") Qubit in the $|-\rangle$ state (-1 eigenvector of $\sigma_x$)
  • "i" (aliases: "Y+", "Yp") Qubit in the $|i\rangle$ state (+1 eigenvector of $\sigma_y$)
  • "-i" (aliases: "Y-", "Ym") Qubit in the $|-i\rangle$ state (+1 eigenvector of $\sigma_y$)

"Qubit" and "S=1/2" Operators

Operators or gates associated with "Qubit" sites can be made using the op function, for example

H = op("H",s)
H3 = op("H",sites[3])

Single-qubit operators:

  • "X" (aliases: "σx", "σ1") Pauli X operator
  • "Y" (aliases: "σy", "σ2") Pauli Y operator
  • "iY" (aliases: "iσy", "iσ2") Pauli Y operator times i
  • "Z" (aliases: "σz", "σ3") Pauli Z operator
  • "√NOT" (aliases: "X")
  • "H" Hadamard gate
  • "Phase" (takes optional argument: ϕ=π/2) (aliases: "P", "S")
  • "π/8" (aliases: "T")
  • "Rx" (takes argument: θ) Rotation around x axis
  • "Ry" (takes argument: θ) Rotation around y axis
  • "Rz" (takes argument: θ) Rotation around z axis
  • "Rn" (takes arguments: θ, ϕ, λ) (aliases: "Rn̂") Rotation about axis n=(θ, ϕ, λ)
  • "Proj0" (aliases: "ProjUp", "projUp") Operator $|0\rangle\langle 0|$
  • "Proj1" (aliases: "ProjDn", "projDn") Operator $|1\rangle\langle 1|$

Spin operators:

  • "Sz" (aliases: "Sᶻ") Spin z operator $S^z = \frac{1}{2} \sigma_z$
  • "S+" (alises: "S⁺", "Splus") Raising operator $S^+ = S^x + iS^y$
  • "S-" (aliases: "S⁻", "Sminus") Lowering operator $S^- = S^x - iS^y$
  • "Sx" (alises: "Sˣ") Spin x operator $S^x = \frac{1}{2} \sigma_x$
  • "iSy" (aliases: "iSʸ") i times spin y operator $iS^y = \frac{i}{2} \sigma_y$
  • "Sy" (aliases: "Sʸ") Spin y operator $S^y = \frac{1}{2} \sigma_y$
  • "S2" (aliases: "S²"`) Square of spin vector operator $S^2=\vec{S}\cdot\vec{S}=\frac{3}{4} I$
  • "ProjUp" (aliases: "projUp", "Proj0") Operator $|\!↑\rangle\langle ↑\!|$
  • "ProjDn" (aliases: "projDn", "Proj1") Operator $|\!↓\rangle\langle ↓\!|$

Two-qubit gates:

  • "CNOT" (aliases: "CX") Controlled NOT gate
  • "CY" Controlled Y gate
  • "CZ" Controlled Z gate
  • "CPHASE" (aliases: "Cphase") Controlled Phase gate
  • "CRx" (aliases: "CRX") (takes arguments: θ)
  • "CRy" (aliases: "CRY") (takes arguments: θ)
  • "CRz" (aliases: "CRZ") (takes arguments: θ)
  • "CRn" (aliases: "CRn̂") (takes arguments: θ, ϕ, λ)
  • "SWAP" (aliases: "Swap")
  • "√SWAP" (aliases: "√Swap")
  • "iSWAP" (aliases: "iSwap")
  • "√iSWAP" (aliases: "√iSwap")
  • "Rxx" (aliases: "RXX") (takes arguments: ϕ) Ising (XX) coupling gate
  • "Ryy" (aliases: "RYY") (takes arguments: ϕ) Ising (YY) coupling gate
  • "Rzz" (aliases: "RZZ") (takes arguments: ϕ) Ising (ZZ) coupling gate

Three-qubit gates:

  • "Toffoli" (aliases "CCNOT", "CCX", "TOFF")
  • "Fredkin" (aliases "CSWAP", "CSwap", "CS")

Four-qubit gates:

  • "CCCNOT"

"S=1" SiteType

Site indices with the "S=1" site type represent $S=1$ spins with the states $|\!\uparrow\rangle$, $|0\rangle$, $|\!\downarrow\rangle$.

Making a single "S=1" site or collection of N "S=1" sites

s = siteind("S=1")
sites = siteinds("S=1",N)

Available keyword arguments for enabling and customizing quantum numbers (QN) subspaces:

  • conserve_qns (default: false): conserve total $S^z$
  • conserve_sz (default: conserve_qns): conserve total $S^z$
  • qnname_sz (default: "Sz"): name of total $S^z$ QN

For example:

sites = siteinds("S=1",N; conserve_sz=true, qnname_sz="TotalSz")

"S=1" States

The available state names for "S=1" sites are:

  • "Up" (aliases: "Z+", "↑") spin in the up state
  • "Z0" (aliases: "0") spin in the Sz=0 state
  • "Dn" (aliases: "Z-", "↓") spin in the Sz=0 state

"S=1" Operators

Operators associated with "S=1" sites can be made using the op function, for example

Sz = op("Sz",s)
Sz4 = op("Sz",sites[4])

Spin operators:

  • "Sz" (aliases: "Sᶻ")
  • "Sz2" Square of S^z operator
  • "S+" (alises: "S⁺", "Splus")
  • "S-" (aliases: "S⁻", "Sminus")
  • "Sx" (alises: "Sˣ")
  • "Sx2" Square of S^x operator
  • "iSy" (aliases: "iSʸ")
  • "Sy" (aliases: "Sʸ")
  • "Sy2" Square of S^y operator
  • "S2" (aliases: "S²"`)

"Boson" SiteType

The "Boson" site type is an alias for the "Qudit" site type. Please see more information about "Qudit" below:

"Qudit" SiteType

Making a single "Qudit" site or collection of N "Qudit" sites

s = siteind("Qudit")
sites = siteinds("Qudit",N)

Available keyword arguments for enabling and customizing quantum numbers (QN) subspaces:

  • dim (default: 2): dimension of the index (number of qudit or boson values)
  • conserve_qns (default: false): conserve total qudit or boson number
  • conserve_number (default: conserve_qns): conserve total qudit or boson number
  • qnname_number (default: "Number"): name of total qudit or boson number QN

For example:

sites = siteinds("Qudit",N; conserve_number=true)

"Qudit" and "Boson" Operators

Operators associated with "Qudit" sites can be made using the op function, for example

A = op("A",s)
A4 = op("A",sites[4])

Single-qudit operators:

  • "A" (aliases: "a")
  • "Adag" (aliases: "adag", "a†")
  • "N" (aliases: "n")

Two-qudit operators:

  • "ab"
  • "a†b"
  • "ab†"
  • "a†b†"

"Fermion" SiteType

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

Making a single "Fermion" site or collection of N "Fermion" sites

s = siteind("Fermion")
sites = siteinds("Fermion",N)

Available keyword arguments for enabling and customizing quantum numbers (QN) subspaces:

  • conserve_qns (default: false): conserve total number of fermions
  • conserve_nf (default: conserve_qns): conserve total number of fermions
  • conserve_nfparity (default: conserve_qns): conserve total fermion number parity
  • qnname_nf (default: "Nf"): name of total fermion number QN
  • qnname_nfparity (default: "NfParity"): name of total fermion number parity QN

For example:

sites = siteinds("Fermion",N; conserve_nfparity=true)

"Fermion" States

The available state names for "Fermion" sites are:

  • "0" (aliases: "Emp") unoccupied fermion site
  • "1" (aliases: "Occ") occupied fermion site

"Fermion" Operators

Operators associated with "Fermion" sites can be made using the op function, for example

C = op("C",s)
C4 = op("C",sites[4])

Single-fermion operators:

  • "N" (aliases: "n") Density operator
  • "C" (aliases: "c") Fermion annihilation operator
  • "Cdag" (aliases: "cdag", "c†") Fermion creation operator
  • "F" Jordan-Wigner string operator

"Electron" SiteType

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

Making a single "Electron" site or collection of N "Electron" sites

s = siteind("Electron")
sites = siteinds("Electron",N)

Available keyword arguments for enabling and customizing quantum numbers (QN) subspaces:

  • conserve_qns (default: false): conserve total number of electrons
  • conserve_sz (default: conserve_qns): conserve total $S^z$
  • conserve_nf (default: conserve_qns): conserve total number of electrons
  • conserve_nfparity (default: conserve_qns): conserve total electron number parity
  • qnname_sz (default: "Sz"): name of total $S^z$ QN
  • qnname_nf (default: "Nf"): name of total electron number QN
  • qnname_nfparity (default: "NfParity"): name of total electron number parity QN

For example:

sites = siteinds("Electron",N; conserve_nfparity=true)

"Electron" States

The available state names for "Electron" sites are:

  • "Emp" (aliases: "0") unoccupied electron site
  • "Up" (aliases: "↑") electron site occupied with one up electron
  • "Dn" (aliases: "↓") electron site occupied with one down electron
  • "UpDn" (aliases: "↑↓") electron site occupied with two electrons (one up, one down)

"Electron" Operators

Operators associated with "Electron" sites can be made using the op function, for example

Cup = op("Cup",s)
Cup4 = op("Cup",sites[4])

Single-fermion operators:

  • "Ntot" (aliases: "ntot") Total density operator
  • "Nup" (aliases: "n↑") Up density operator
  • "Ndn" (aliases: "n↓") Down density operator
  • "Cup" (aliases: "c↑") Up-spin annihilation operator
  • "Cdn" (aliases: "c↓") Down-spin annihilation operator
  • "Cdagup" (aliases: "c†↑") Up-spin creation operator
  • "Cdagdn" (aliases: "c†↓") Down-spin creation operator
  • "Sz" (aliases: "Sᶻ")
  • "Sx" (aliases: "Sˣ")
  • "S+" (aliases: "Sp", "S⁺","Splus")
  • "S-" (aliases: "Sm", "S⁻", "Sminus")
  • "F" Jordan-Wigner string operator
  • "Fup" (aliases: "F↑") Up-spin Jordan-Wigner string operator
  • "Fdn" (aliases: "F↓") Down-spin Jordan-Wigner string operator

Non-fermionic single particle operators (these do not have Jordan-Wigner string attached, so will commute within systems such as OpSum or the apply function):

  • "Aup" (aliases: "a↑") Up-spin annihilation operator
  • "Adn" (aliases: "a↓") Down-spin annihilation operator
  • "Adagup" (aliases: "a†↑") Up-spin creation operator
  • "Adagdn" (aliases: "a†↓") Down-spin creation operator

"tJ" SiteType

"tJ" sites are similar to electron sites, but cannot be doubly occupied The states of site indices with the "tJ" SiteType correspond to $|0\rangle$, $|\!\uparrow\rangle$, $|\!\downarrow\rangle$.

Making a single "tJ" site or collection of N "tJ" sites

s = siteind("tJ")
sites = siteinds("tJ",N)

Available keyword arguments for enabling and customizing quantum numbers (QN) subspaces:

  • conserve_qns (default: false): conserve total number of fermions
  • conserve_nf (default: conserve_qns): conserve total number of fermions
  • conserve_nfparity (default: conserve_qns): conserve total fermion number parity
  • qnname_nf (default: "Nf"): name of total fermion number QN
  • qnname_nfparity (default: "NfParity"): name of total fermion number parity QN

For example:

sites = siteinds("tJ",N; conserve_nfparity=true)

"tJ" States

The available state names for "tJ" sites are:

  • "Emp" (aliases: "0") unoccupied site
  • "Up" (aliases: "↑") site occupied with one up electron
  • "Dn" (aliases: "↓") site occupied with one down electron

"tJ" Operators

Operators associated with "tJ" sites can be made using the op function, for example

Cup = op("Cup",s)
Cup4 = op("Cup",sites[4])

Single-fermion operators:

  • "Ntot" (aliases: "ntot") Total density operator
  • "Nup" (aliases: "n↑") Up density operator
  • "Ndn" (aliases: "n↓") Down density operator
  • "Cup" (aliases: "c↑") Up-spin annihilation operator
  • "Cdn" (aliases: "c↓") Down-spin annihilation operator
  • "Cdagup" (aliases: "c†↑") Up-spin creation operator
  • "Cdagdn" (aliases: "c†↓") Down-spin creation operator
  • "Sz" (aliases: "Sᶻ")
  • "Sx" (aliases: "Sˣ")
  • "S+" (aliases: "Sp", "S⁺","Splus")
  • "S-" (aliases: "Sm", "S⁻", "Sminus")
  • "F" Jordan-Wigner string operator
  • "Fup" (aliases: "F↑") Up-spin Jordan-Wigner string operator
  • "Fdn" (aliases: "F↓") Down-spin Jordan-Wigner string operator

Non-fermionic single particle operators (these do not have Jordan-Wigner string attached, so will commute within systems such as OpSum or the apply function):

  • "Aup" (aliases: "a↑") Up-spin annihilation operator
  • "Adn" (aliases: "a↓") Down-spin annihilation operator
  • "Adagup" (aliases: "a†↑") Up-spin creation operator
  • "Adagdn" (aliases: "a†↓") Down-spin creation operator