Site Basis

Basis states of a single site are described by one or more quantum numbers as in:

<SITEBASIS name="hardcore boson">
<QUANTUMNUMBER name="N" min="0" max="1"/>
</SITEBASIS>

<SITEBASIS name="spin-1/2">
<QUANTUMNUMBER name="S" min="1/2" max="1/2"/>
<QUANTUMNUMBER name="Sz" min="-1/2" max="1/2"/>
</SITEBASIS>

<SITEBASIS name="fermion">
<QUANTUMNUMBER name="Nup" min="0" max="1" type="fermionic"/>
<QUANTUMNUMBER name="Ndown" min="0" max="1" type="fermionic"/>
</SITEBASIS>

The first example above describes a hardcore boson (one bosonic mode with occupation restricted to 0 or 1), the second a spin-1/2, and the third a spinful fermion that allows double occupancy (independent Nup and Ndown, each ranging over 0 or 1).

Note that the spin-1/2 example above declares both S and Sz, even though Sz alone would already distinguish the two states: the total spin S is needed separately because it appears in the matrix elements of the spin operators defined in Quantum Operators.

The <SITEBASIS> takes a name attribute by which it can later be referenced. The <QUANTUMNUMBER> elements each take a name attribute, and minimum and maximum values via the min and max attributes. The quantum numbers can take values between min, min+1, min+2 … up to max. Optionally, a type attribute can be set to bosonic (the default) or fermionic. It should be set to fermionic when the quantum number counts a fermionic degree of freedom (e.g. a fermion occupation number), so that operators built from it anticommute correctly with fermionic operators on other sites.

The range of the quantum numbers can be parametrized by input parameters, and a default can be specified such as in

<SITEBASIS name="boson">
<PARAMETER name="Nmax" default="infinity"/>
<QUANTUMNUMBER name="N" min="0" max="Nmax"/>
</SITEBASIS>

For more complicated models, such as a t-J model, sometimes several equivalent choices of quantum numbers are possible. We can label the states either by the particle number and spin, or by the number of up and down spins:

<SITEBASIS name="t-J">
<QUANTUMNUMBER name="N" min="0" max="1" type="fermionic"/>
<QUANTUMNUMBER name="S" min="N/2" max="N/2"/>
<QUANTUMNUMBER name="Sz" min="-S" max="S"/>
</SITEBASIS>

<SITEBASIS name="alternative t-J">
<QUANTUMNUMBER name="Nup" min="0" max="1" type="fermionic"/>
<QUANTUMNUMBER name="Ndown" min="0" max="1-Nup" type="fermionic"/>
</SITEBASIS>

In the first version, min="N/2" max="N/2" pins S to exactly N/2 rather than letting it range freely: an empty site (N=0) is forced to have S=0, and a singly occupied site (N=1) is forced to have S=1/2, which is exactly the no-double-occupancy constraint that defines the t-J model. The two site bases describe the same three physical states (empty, spin up, spin down) and are equally valid; which one is more convenient depends on the Hamiltonian: the N,S basis is natural when the total spin is a useful quantum number, while the Nup,Ndown basis is often simpler when writing operators that act separately on the up- and down-spin fermions, such as the cdag_up/c_up operators in the fermion_hop example in Quantum Operators.

For an overview of the rest of this section, see ALPS Model Definitions. For how single-site bases are combined into the basis of the whole lattice, see Lattice Basis. For the other ALPS documentation sections, see the General Introduction.