Using SeQuant¶

SeQuant is a general-purpose symbolic tensor algebra, but the primary use case is in quantum many-body physics. The following is a brief tutorial on using SeQuant for this purpose.

Wick’s Theorem¶

To get started let’s use SeQuant to apply Wick’s theorem to a simple product of elementary (creation and annihilation) fermionic operators:

\[a_{p_3} a_{p_4} a^\dagger_{p_1} a^\dagger_{p_2}.\]

This is achieved by the following SeQuant program:

Index p1(L"p_1"), p2(L"p_2"), p3(L"p_3"), p4(L"p_4");

auto cp1 = fcrex(p1), cp2 = fcrex(p2);
auto ap3 = fannx(p3), ap4 = fannx(p4);

std::wcout << to_latex(ap3 * ap4 * cp1 * cp2) << " = "
           << to_latex(FWickTheorem{ap3 * ap4 * cp1 * cp2}
                           .full_contractions(false)
                           .compute())
           << std::endl;

Hint

All core user-facing SeQuant code lives in C++ namespace sequant, and the shown code assumes this namespace has been imported via using namespace sequant.

Running this program should produce a LaTeX expression for this formula:

\[{{a_ {{p_3}}}{a_ {{p_4}}}{a^{{p_1}}}{a^{{p_2}}}} = \bigl( - {{a^{{p_1}{p_2}}_ {{p_3}{p_4}}}} - {{s^{{p_1}}_ {{p_3}}}{s^{{p_2}}_ {{p_4}}}} + {{s^{{p_1}}_ {{p_3}}}{a^{{p_2}}_ {{p_4}}}} - {{s^{{p_1}}_ {{p_4}}}{a^{{p_2}}_ {{p_3}}}} + {{s^{{p_2}}_ {{p_3}}}{s^{{p_1}}_ {{p_4}}}} - {{s^{{p_2}}_ {{p_3}}}{a^{{p_1}}_ {{p_4}}}} + {{s^{{p_2}}_ {{p_4}}}{a^{{p_1}}_ {{p_3}}}}\bigr)\]

where the tensor notation is used to denote elementary and composite normal-ordered (or, shortly, normal) operators:

\[a^p \equiv a_p^\dagger\]

\[a^{p_1 p_2 \dots p_c}_ {q_1 q_2 \dots q_a} \equiv a_ {p_1}^\dagger a_ {p_2}^\dagger \dots a_ {p_c}^\dagger a_ {q_a} \dots a_ {q_2} a_ {q_1}\]

\(s^p_q \equiv \langle q | p \rangle\) denotes inner products (“overlaps”) of 1-particle states. Wick’s theorem can of course be applied directly to products of normal composite operators, e.g,

auto nop1 = ex<FNOperator>(cre(p1, p2), ann(p3, p4));
// OR
// auto nop1 = ex<FNOperator>(cre({p1, p2}), ann({p3, p4}));
// OR
// auto nop1 = ex<FNOperator>(cre(std::vector{p1, p2}),
//                            ann(std::array{L"p3", L"p4"}));
// OR
// auto nop1 = ex<FNOperator>(cre(std::set{"p1", "p2"}),
//                            ann(std::vector{L"p3", L"p4"}));
auto nop2 = ex<FNOperator>(cre({L"p_5"}), ann({L"p_6", L"p_7"}));

std::wcout
    << to_latex(nop1 * nop2) << " = "
    << to_latex(FWickTheorem{nop1 * nop2}.full_contractions(false).compute())
    << std::endl;

produces

\[{{a^{{p_1}{p_2}}_ {{p_3}{p_4}}}{a^{␣\,{p_5}}_ {{p_6}{p_7}}}} = \bigl({a^{␣\,{p_1}{p_2}{p_5}}_ {{p_3}{p_4}{p_6}{p_7}}} - {{s^{{p_5}}_ {{p_4}}}{a^{␣\,{p_1}{p_2}}_ {{p_3}{p_6}{p_7}}}} + {{s^{{p_5}}_ {{p_3}}}{a^{␣\,{p_1}{p_2}}_ {{p_4}{p_6}{p_7}}}}\bigr)\]

where \(␣\) is used in number-nonconserving operators to point out the empty “slots”.

Same algebra can be performed for bosons:

auto nop3 = ex<BNOperator>(cre({p1, p2}), ann({p3, p4}));
auto nop4 = ex<BNOperator>(cre({L"p_5", L"p_6"}), ann({L"p_7"}));

std::wcout
    << to_latex(nop3 * nop4) << " = "
    << to_latex(BWickTheorem{nop3 * nop4}.full_contractions(false).compute())
    << std::endl;

\[{{b^{{p_1}{p_2}}_ {{p_3}{p_4}}}{b^{{p_5}{p_6}}_ {␣\,{p_7}}}} = \bigl( {b^{{p_5}{p_1}{p_2}{p_6}}_ {␣\,{p_3}{p_4}{p_7}}} + {{s^{{p_6}}_ {{p_3}}}{b^{{p_5}{p_2}{p_1}}_{␣\,{p_4}{p_7}}}} + {{s^{{p_5}}_{{p_3}}}{b^{{p_1}{p_2}{p_6}}_{␣\,{p_4}{p_7}}}} + {{s^{{p_6}}_{{p_4}}}{b^{{p_5}{p_1}{p_2}}_{␣\,{p_3}{p_7}}}} + {{s^{{p_5}}_{{p_4}}}{b^{{p_2}{p_1}{p_6}}_{␣\,{p_3}{p_7}}}} + {{s^{{p_5}}_{{p_3}}}{s^{{p_6}}_{{p_4}}}{b^{{p_1}{p_2}}_{␣\,{p_7}}}} + {{s^{{p_6}}_{{p_3}}}{s^{{p_5}}_{{p_4}}}{b^{{p_2}{p_1}}_{␣\,{p_7}}}} \bigr)\]

where \(b\) denotes normal bosonic operators constructed analogously with the normal fermionic operators \(a\)

Register Index Spaces¶

Tensor expressions annotated by abstract indices are common. In some contexts all tensor modes refer to the same range or underlying vector space (as in all examples shown so far); then there is no need to distinguish modes of different types. But in some contexts indices carry important semantic meaning. For example, the energy expression in the coupled-cluster method,

\[E_\mathrm{CC} = F^{a_1}_ {i_1} t^{i_1}_ {a_1} + \frac{1}{4} \bar{g}^{a_1 a_2}_ {i_1 i_2} (t^{i_1 i_2}_ {a_1 a_2} + 2 t^{i_1}_ {a_1} t^{i_2}_ {a_2})\]

contains tensors with 2 types of modes, denoted by \(i\) and \(a\), that represent single-particle (SP) states occupied and unoccupied in the reference state, respectively. To simplify symbolic manipulation of such expressions SeQuant allows to define a custom vocabulary of index spaces and to define their set-theoretic relationships. The following example illustrates the full space denoted by \(p\) partitioned into occupied \(i\) and unoccupied \(a\) base subspaces: