Hamiltonian Descriptions

With these elements we can now describe the Hamiltonian of a model. A simple hardcore boson model could be:

<HAMILTONIAN name="hardcore boson">
<PARAMETER name="mu" default="0"/>
<PARAMETER name="t" default="1"/>
<PARAMETER name="t'" default="1"/>
<BASIS ref="hardcore boson"/>
<SITETERM type="0">
    -mu*n
</SITETERM>
<BONDTERM type="0" source="i" target="j">
    -t*(bdag(i)*b(j)+bdag(j)*b(i))
</BONDTERM>
<BONDTERM type="1" source="i" target="j">
    -t'*(bdag(i)*b(j)+bdag(j)*b(i))
</BONDTERM>
</HAMILTONIAN>

First, default values can be specified for parameters such as coupling constants by using <PARAMETER> elements. Next, one <BASIS> element specifies the basis used for the model, either fully specified inline or by a reference (using the ref attribute).

The terms of the Hamiltonian are next specified by site terms, associated with the sites of the lattice, and bond terms, associated with the bonds. Each of the <SITETERM> and <BONDTERM> elements can optionally take a type attribute, specifying which type of site (or bond) the term applies to — the same types specified in the lattice description. Omitting the type attribute applies the term to all sites or bonds for which no other term is explicitly specified.

The <SITETERM> elements contain terms of the Hamiltonian associated with a single site. In the above example the term mu refers to the parameter mu while the term n refers to the operator n described in Quantum Operators. In the <BONDTERM> elements, the operators must refer to two different sites; this is done by adding the site index in parentheses after the operator, e.g. as in n(i) to act on site i. The source and target attributes name the variables used to refer to the two sites (i and j in the example).

To simplify writing Hamiltonians, previously defined site and bond operators can be reused instead of writing out matrix elements again. For a transverse field spin model we can use the Sx and exchange operators defined in Quantum Operators:

<HAMILTONIAN name="spin">
<PARAMETER name="J" default="1"/>
<PARAMETER name="h" default="0"/>
<PARAMETER name="Gamma" default="0"/>
<BASIS ref="spin"/>
<SITETERM site="i">
    -h*Sz(i)-Gamma*Sx(i)
</SITETERM>
<BONDTERM source="i" target="j">
    J*exchange(i,j)
</BONDTERM>
</HAMILTONIAN>

Site-type dependent coupling terms can either be specified as in the first example, giving a type attribute restricting the applicability of a site or bond term, or by using the # wildcard character in the name of coupling constants, which will be replaced by the type of the site or bond as in:

<HAMILTONIAN name="spin">
<PARAMETER name="J" default="1"/>
<PARAMETER name="h" default="0"/>
<PARAMETER name="Gamma" default="0"/>
<BASIS ref="spin"/>
<SITETERM site="i">
    <PARAMETER name="h#" default="h"/>
    <PARAMETER name="Gamma#" default="Gamma"/>
    -h#*Sz(i)-Gamma#*Sx(i)
</SITETERM>
<BONDTERM source="i" target="j">
    <PARAMETER name="J#" default="J"/>
    J#*exchange(i,j)
</BONDTERM>
</HAMILTONIAN>

We can now specify h0 and Gamma0 on sites of type 0 and J0 on bonds of type 0, h1, Gamma1, and J1 for type 1, and so on — this is exactly how, e.g., the anisotropic2d unit cell (see A Library of Lattices and Graphs) lets a model assign different couplings J0/J1 to bonds along each lattice direction. J, h, and Gamma here are the same parameters described in ALPS Model Definitions.

Extensions to more complex terms, such as 3- and 4-site terms, are in preparation and will be included here as soon as the ALPS libraries support such terms.

For an overview of the rest of this section, see ALPS Model Definitions. For the operators used in these Hamiltonian terms, see Quantum Operators. For the other ALPS documentation sections, see the General Introduction.