Position-Dependent Couplings

On an ordinary ALPS lattice, every coupling in a model (mu, t, J, h, …) has one value, used on every site or every bond of a given type. Many problems need couplings that change from place to place: a harmonic trap or a linear tilt in the chemical potential, a field gradient, or a random potential. ALPS handles these with two pieces:

1. an inhomogeneous lattice, which tells ALPS to evaluate the site terms (or bond terms) separately on every site (or bond), and
2. a parameter expression that uses the coordinates x, y and z of that site or bond, or a random number.

Nothing in the model has to change: the built-in models in models.xml can be used as they are.

A first example

The following parameter file puts hardcore bosons in a linear potential $\mu(x) = \mu_\text{left} - s\,x$ on a chain of 6 sites:

LATTICE="inhomogeneous chain lattice"
L=6
MODEL="hardcore boson"
t=1
mu_left=1
slope=0.4
mu="mu_left - slope*x"
T=1
THERMALIZATION=5000
SWEEPS=20000
MEASURE[Local Density]=1

Run it as usual, for example

parameter2xml parm_tilt
dirloop_sse parm_tilt.in.xml

The Local Density measurement then shows the density falling from the left end of the chain ($x=0$) to the right end ($x=5$).

The value of mu is quoted because it is an expression rather than a number. It is evaluated once for every site, with x set to that site’s coordinate. Expressions can refer to other parameters, here mu_left and slope.

Inhomogeneous lattices

A lattice is inhomogeneous when its <LATTICEGRAPH> contains an <INHOMOGENEOUS> element. The lattice library lattices.xml already contains four of them:

All four have open boundaries, and only their site terms are position dependent.

To make your own lattice inhomogeneous, add the element after <UNITCELL>:

<LATTICEGRAPH name = "inhomogeneous periodic chain lattice">
  <FINITELATTICE>
    <LATTICE ref="chain lattice"/>
    <EXTENT dimension="1" size="L"/>
    <BOUNDARY type="periodic"/>
  </FINITELATTICE>
  <UNITCELL ref="simple1d"/>
  <INHOMOGENEOUS><VERTEX/><EDGE/></INHOMOGENEOUS>
</LATTICEGRAPH>

<VERTEX/> makes the site terms position dependent and <EDGE/> makes the bond terms position dependent; use either or both. They apply to all sites or all bonds: restricting them to one type with a type attribute is not supported. The Bose glass page uses the same element to give the square lattice periodic boundaries.

Coordinates in expressions

On an inhomogeneous lattice, an expression can use these variables:

The coordinates are the ones printgraph prints for the lattice. For the chain, square and simple cubic lattices they are the integer cell positions, from 0 to L-1 along the first direction. A trap centred on an L$\times$L square lattice is therefore

LATTICE="inhomogeneous square lattice"
L=10
mu="4 - 0.2*((x-(L-1)/2)^2 + (y-(L-1)/2)^2)"

Bond terms use the midpoint of the bond. On a chain, the bond between sites $x=2$ and $x=3$ sees x=2.5, so with <EDGE/> in the lattice

J="1 + 0.2*x"

gives that bond $J = 1.5$.

The same variables can also be used inside a model definition. For example, this <HAMILTONIAN> builds the tilt into the model, so the parameter file only sets mu and slope:

<HAMILTONIAN name="tilted hardcore boson">
  <PARAMETER name="mu" default="0"/>
  <PARAMETER name="slope" default="0"/>
  <PARAMETER name="t" default="1"/>
  <BASIS ref="hardcore boson"/>
  <SITETERM site="i">-(mu - slope*x)*n(i)</SITETERM>
  <BONDTERM source="i" target="j">-t*(bdag(i)*b(j)+bdag(j)*b(i))</BONDTERM>
</HAMILTONIAN>

As with LATTICE_LIBRARY, a file given with MODEL_LIBRARY replaces the built-in models.xml, so add the new Hamiltonian to a copy of it.

Two errors are common:

• On a lattice without <INHOMOGENEOUS>, the coordinates are not defined, and the program stops with Cannot evaluate expression ... followed by the expression that uses them.
• x, y and z are reserved on an inhomogeneous lattice. Defining one of them as a parameter stops the program with x, y or z is predefined as parameter and used as coordinate.

Functions and operators

Expressions can use +, -, *, /, ^ (power) and parentheses, the constant Pi (also PI or pi), and these functions:

normal_random is another name for gaussian_random.

Random disorder

On an inhomogeneous lattice, a random function is evaluated separately on every site (or bond), so

mu="delta*(2*random() - 1)"

gives every site its own chemical potential, uniformly distributed between -delta and delta.

The random numbers come from their own generator, seeded with DISORDERSEED (default 0). This seed is separate from SEED, the seed of the Monte Carlo updates. Runs with the same DISORDERSEED therefore see the same disorder, even when their SEED differs. To average over disorder, give each realization its own DISORDERSEED in a { } block, which makes it a separate task:

{ DISORDERSEED = 34275; }
{ DISORDERSEED = 49802; }
{ DISORDERSEED = 82529; }

and average the results of the tasks afterwards. The Bose glass page has a complete example.

Which programs support it

Position-dependent couplings work in the exact diagonalization codes (sparsediag, fulldiag), in dmrg, and in the quantum Monte Carlo codes (dirloop_sse, loop, worm). The optical lattice page uses dirloop_sse to simulate bosons in a three-dimensional trap.

worm needs a lattice, not a graph

A GRAPH from the lattice library (a list of vertices and edges, without a unit cell) can be used with dirloop_sse and the exact diagonalization codes, but not with worm, which stops with

Cannot save a measurement of size 0.

To run worm on a small cluster such as a single triangle, describe the cluster as the unit cell of a lattice with open boundaries and set L=1. The Kagome unit cell in lattices.xml is a triangle of 3 sites, so

<LATTICEGRAPH name = "open kagome lattice">
  <FINITELATTICE>
    <LATTICE ref="triangular lattice"/>
    <PARAMETER name="W" default="L"/>
    <EXTENT dimension="1" size="L"/>
    <EXTENT dimension="2" size="W"/>
    <BOUNDARY type="open"/>
  </FINITELATTICE>
  <UNITCELL ref="Kagome"/>
</LATTICEGRAPH>

with LATTICE="open kagome lattice" and L=1 is a triangle that worm accepts. Add <INHOMOGENEOUS><VERTEX/></INHOMOGENEOUS> to it to make the couplings position dependent as well.

For the Hamiltonian terms these parameters enter, see Hamiltonian Descriptions. For how lattices and unit cells are defined, see Lattice Definitions. For the other parameters common to ALPS programs, see Common Parameters.