  Periodic field patterns of atoms and their charges/spins/orbits emerge in
crystals, forming novel states of matter called emergent crystals (ECs). In
recent years, they are observed in diverse systems such as skyrmion crystals in
helimagnets, and periodic ripples in 2D materials. ECs essentially changes the
properties of material underneath, and are deformable when subject to various
effective fields. A major challenge in application is first to predict what
kind of EC will appear in the system of interest, and how to quantify its
`elasticity' when subject to an effective field. Here we establish the
theoretical framework of thermodynamics for deformable ECs, and derive from it
the linear constitutive equations when subject to the primary external field.
We provide a systematic study on the ECs that may appear in helimagnets induced
by the Dzyaloshinskii-Moriya interaction, and analyze their elasticity when
subject to bias magnetic fields. We construct in this work the basis of
emergent elasticity, a new branch studying deformable emergent crystals under
effective fields.
