Department of Physics, University of Washington,
Seattle, Washington 98195-1560
We show that disordered flat phases in crystal surfaces
are equivalent to valence bond type phases
in integer and half-integer spin quantum chains.
In the quantum spin representation the
disordered flat phase represents a
fluid type phase with long-range antiferromagnetic spin order.
This order is stabilized dynamically by the hopping of the particles
and short-range spin exchange interactions.
The mass of Neel solitons is finite.
Numerical finite-size-scaling results confirm this.
We identify the order parameter of the valence bond phase.
The Haldane conjecture suggests a fundamental difference between
half-integer and integer antiferromagnetic Heisenberg spin chains.
We find that disordered flat phases are realized in both cases,
have exactly the same type of long-range antiferromagnetic spin order,
and are stabilized by exactly the same mechanism.
They differ only in the mathematical formulation of
broken symmetry in the spin representation.
We suggest experimental methods of observing disordered flat phases
in crystal surfaces.