  Very thin elastic sheets, even at zero temperature, exhibit nonlinear elastic
response by virtue of their dominant bending modes. Their behavior is even
richer at finite temperature. Here we use molecular dynamics (MD) to study the
vibrations of a thermally fluctuating two-dimensional elastic sheet with one
end clamped at its zero-temperature length. We uncover a tilt phase in which
the sheet fluctuates about a mean plane inclined with respect to the
horizontal, thus breaking reflection symmetry. We determine the phase behavior
as a function of the aspect ratio of the sheet and the temperature. We show
that tilt may be viewed as a type of transverse buckling instability induced by
clamping coupled to thermal fluctuations and develop an analytic model that
predicts the tilted and untilted regions of the phase diagram. Qualitative
agreement is found with the MD simulations. Unusual response driven by control
of purely geometric quantities like the aspect ratio, as opposed to external
fields, offers a very rich playground for two-dimensional mechanical
metamaterials.
