vev

◼ vev[expr] returns the vacuum expectation value (vev) of an operator expression.
◼ vevwick[expr] calculates the vacuum expectation value of an operator expression using Wick's theorem.

◼ vev and vevwick take into account the type of the vacuum associated with a group of operators as defined by ordering.

◼ It is assumed that the vacuum state has well defined number of particles. This implies that the vev of an odd number of operators is zero by definition.

◼ The vev of a canonically normal ordered product of operators is zero.

◼ By definition, vev[nc[conj[VACUUM],VACUUM]]=1.

◼ See also: ordering, normalorder, normalorderwick.

Further examples

Load sneg and declare some quantities.

<< sneg`sneg`;
snegfermionoperators[c, d]
ordering[c] = SEA;
ordering[d] = EMPTY;
snegfreeindexes[k, σ]

Simple cases: odd number of operators or an annihilation operator applied on the vacuum state.

vev[c[CR]] == 0
vev[c[AN]] == 0
vev[d[CR]] == 0
vev[d[AN]] == 0
vev[nc[c[CR, 1], c[CR, 2], c[CR, 3]]] == 
      0
vev[nc[d[CR, 1], d[CR, 2], d[CR, 3]]] == 
      0
vev[nc[expr, d[AN]]] == 0
vev[nc[d[CR], expr]] == 0

In the case of the Fermi sea vacuum state, information about the true annihilation/creation nature of an operators is required.

vev[nc[expr, c[AN, k, σ]]]
snegAssuming[k /: k > 0 = True, %] == 
      0
vev[nc[expr, c[CR, k, σ]]]
snegAssuming[k /: k < 0 = True, %] == 
      0
vev[nc[c[CR, k, σ], expr]]
snegAssuming[k /: k > 0 = True, %] == 
      0
vev[nc[c[AN, k, σ], expr]]
snegAssuming[k /: k < 0 = True, %] == 0

vev of a sum is defined to be a sum of the vacuum expectation values of the addends.

SetAttributes[eps, NumericFunction]
Conjugate[eps[k_]] ^= eps[k];
H = 
      sum[eps[k] number[
            c[k], σ], {k, σ}]
vev[H]