A Library of Lattices and Graphs

Since similar lattices, finite lattices, and graphs appear again and again in many models and simulations, it is advantageous to define the commonly used ones once, in a “library,” and refer to them by name — for example, as an input parameter to a simulation. The first step is to divide the definition of a <LATTICEGRAPH> into four parts:

<LATTICE name="square" dimension="2">
<BASIS>
    <VECTOR> 1 0 </VECTOR>
    <VECTOR> 0 1 </VECTOR>
</BASIS>
</LATTICE>

<FINITELATTICE name="rectangular periodic" dimension="2">
<LATTICE ref="square"/>
<PARAMETER name="L" />
<PARAMETER name="W" default="L" />
<EXTENT dimension="1" size="L"/>
<EXTENT dimension="2" size="W"/>
<BOUNDARY type="periodic"/>
</FINITELATTICE>

<UNITCELL name="complex example" dimension="2" vertices="2">
<VERTEX id="1" type="0"><COORDINATE> 0.3 0.7 </COORDINATE></VERTEX>
<VERTEX id="2" type="1"><COORDINATE> 0.6 0.3 </COORDINATE></VERTEX>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="1 0"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="0 1"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="2"/></EDGE>
</UNITCELL>
<LATTICEGRAPH>
<FINITELATTICE ref="rectangular periodic"/>
<UNITCELL ref="complex example"/>
</LATTICEGRAPH>

Here we made use of previously defined, named lattices and unit cells (e.g. in a library), which we can then combine by referencing them in the <LATTICEGRAPH> element. Now we can define a <LATTICES> element, which can contain any number of <LATTICE>, <FINITELATTICE>, <UNITCELL>, <GRAPH>, and <LATTICEGRAPH> elements. For example:

<LATTICES>
<LATTICE name="chain lattice" dimension="1">
<BASIS><VECTOR>1</VECTOR></BASIS>
</LATTICE>
<LATTICE name="square lattice" dimension="2">
<PARAMETER name="a" default="1"/>
<BASIS><VECTOR>a 0</VECTOR><VECTOR>0 a</VECTOR></BASIS>
</LATTICE>
<LATTICE name="simple cubic lattice" dimension="3">
<PARAMETER name="a" default="1"/>
<BASIS>
    <VECTOR>a 0 0</VECTOR>
    <VECTOR>0 a 0</VECTOR>
    <VECTOR>0 0 a</VECTOR>
</BASIS>
</LATTICE>
<UNITCELL name="simple1d" dimension="1">
<VERTEX/>
<EDGE><SOURCE vertex="1" offset="0"/><TARGET vertex="1" offset="1"/></EDGE>
</UNITCELL>
<UNITCELL name="simple2d" dimension="2">
<VERTEX/>
<EDGE><SOURCE vertex="1" offset="0 0"/><TARGET vertex="1" offset="0 1"/></EDGE>
<EDGE><SOURCE vertex="1" offset="0 0"/><TARGET vertex="1" offset="1 0"/></EDGE>
</UNITCELL>
<UNITCELL name="anisotropic2d" dimension="2">
<VERTEX/>
<EDGE type="0"><SOURCE vertex="1" offset="0 0"/><TARGET vertex="1" offset="1 0"/></EDGE>
<EDGE type="1"><SOURCE vertex="1" offset="0 0"/><TARGET vertex="1" offset="0 1"/></EDGE>
</UNITCELL>
<UNITCELL name="simple3d" dimension="3" vertices="1">
<VERTEX/>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="1 0 0"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="0 1 0"/></EDGE>
<EDGE><SOURCE vertex="1"/><TARGET vertex="1" offset="0 0 1"/></EDGE>
</UNITCELL>
<LATTICEGRAPH name = "square lattice 3x3">
<FINITELATTICE>
    <LATTICE ref="square lattice"/>
    <EXTENT dimension="1" size="3"/>
    <EXTENT dimension="2" size="3"/>
    <BOUNDARY dimension="1" type="periodic"/>
    <BOUNDARY dimension="2" type="open"/>
</FINITELATTICE>
<UNITCELL ref="simple2d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "dimer">
<FINITELATTICE>
    <LATTICE ref="chain lattice"/>
    <EXTENT dimension="1" size="2"/>
    <BOUNDARY type="open"/>
</FINITELATTICE>
<UNITCELL ref="simple1d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "simple cubic lattice">
<FINITELATTICE>
    <LATTICE ref="simple cubic lattice"/>
    <PARAMETER name="W" default="L"/>
    <PARAMETER name="H" default="W"/>
    <EXTENT dimension="1" size="L"/>
    <EXTENT dimension="2" size="W"/>
    <EXTENT dimension="3" size="H"/>
    <BOUNDARY type="periodic"/>
</FINITELATTICE>
<UNITCELL ref="simple3d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "chain lattice">
<FINITELATTICE>
    <LATTICE ref="chain lattice"/>
    <EXTENT dimension="1" size ="L"/>
    <BOUNDARY type="periodic"/>
</FINITELATTICE>
<UNITCELL ref="simple1d"/>
</LATTICEGRAPH>
<LATTICEGRAPH name = "anisotropic square lattice">
<FINITELATTICE>
    <LATTICE ref="square lattice"/>
    <PARAMETER name="W" default="L"/>
    <EXTENT dimension="1" size="L"/>
    <EXTENT dimension="2" size="W"/>
    <BOUNDARY type="periodic"/>
</FINITELATTICE>
<UNITCELL ref="anisotropic2d"/>
</LATTICEGRAPH>
</LATTICES>

The anisotropic2d unit cell mirrors simple2d, but gives the two bond directions different edge types (0 and 1), so that a model can assign a different coupling to each — for example, J0 along the first lattice direction and J1 along the second.

For an overview of the rest of this section, see Lattice Definitions. For the LATTICE/GRAPH input parameters used to select a lattice or graph by name in a simulation, see Common Parameters. For the other ALPS documentation sections, see the General Introduction.

Lattices and Graphs Check Lattice Graph