Inhomogeneous Coulomb Models Without Defects are Sick: Domain Wall Energies Scaling as Volume (not Boundary Area)

Abstract

In this work we show that coulomb models, ones that obey a $\nabla\cdot\left\langle \mathbf{B}\right\rangle =0$ divergence free constraint - for $\left\langle \mathbf{B}\right\rangle $ being some coarse grained variables related to the microscopic degrees of freedom of the lattice system - are sick without the inclusion of defects with $\nabla\cdot\mathbf{B}\neq0$. We show that for generic boundaries (ones where the fluxes of the pseudo-magnetic fields on the boundary do not cancel: $\left(\left\langle \mathbf{B}_{R}\right\rangle -\left\langle \mathbf{B}_{L}\right\rangle \right)\cdot\mathbf{n}\neq0$ - here $\mathbf{n}$ is the unit normal and $\left\langle \mathbf{B}_{R/L}\right\rangle $ are the two ground state pseudo-magnetic fields on either side of the domain wall) without the inclusion of defects, sharp domain walls (on the order of the width of a unit cell) between different phases of the system cost energy proportional to system size (not boundary area). We present several different examples of this phenomena in the square lattice six vertex model, in quantum dimers and in classical spin ice in the presence of magnetic fields. We also show by example that the condition $\left(\left\langle \mathbf{B}_{R}\right\rangle -\left\langle \mathbf{B}_{L}\right\rangle \right)\cdot\mathbf{n}=0$ is a necessary but not sufficient condition for the boundaries to be compatible - that is domain wall energy to scale with domain wall area and not system size. To further present the importance of boundary conditions in Coulomb systems we show that system boundaries, even ones that satisfy $\int_{\partial V}\mathbf{B}\cdot\mathbf{n}=0$, have thermodynamic consequences - that is there is a cost, extensive in system size, to the Helmholtz free energy.