Figure 2
Extracting universal behavior and dynamical critical exponents from density profiles of the trapped one-dimensional Bose-Hubbard model. (a) Exact density profiles of the one-dimensional harmonically trapped hard-core Bose-Hubbard model for
particles at temperatures
, with larger temperatures corresponding to lower central density. Here,
is the radial displacement in the trap and
is the lattice spacing. These density profiles are nonuniversal; for example, they depend on temperature. (b) Our construction for obtaining universal scaling curves applied to this system, plotting
versus
(defining
,
, and
) for this
,
transition and temperatures
(from closest to the shaded grey band to farthest). The compressibility is approximated by
where
is the trap frequency, and
is obtained by numerically differentiating the density. Lower temperatures display a larger region of collapse. We observe good collapse up to
, and see that the analysis accurately reproduces the homogeneous infinite system’s scaling curve (shaded gray line) within
for
for the transition near
(
) and for
for the transition near
(
). This collapse occurs even for drastically different density profiles obtained by adjusting the trap depth in place of temperature (not shown). With moderately larger particle numbers (
, not shown), the simulated data at low temperatures even more accurately reproduces the infinite homogeneous system’s universal scaling function (the extracted curve lies within the shaded gray region).
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