Entropic Order Superfluid

We explore entropic order, a mechanism by which ordered phases of matter can persist to arbitrarily high temperatures ($T \to \infty$). This counterintuitive effect arises when ordering a subset of degrees of freedom enhances the entropy of the remaining ones, making the ordered state entropically favorable. Interacting bosonic degrees of freedom play a key role, as their unbounded fluctuations circumvent conventional no-go theorems that would otherwise forbid high-temperature order.We present a classical many-body model of interacting ``soft rotors,'' described by radial ($r_i$) and angular ($\theta_i$) coordinates, which exhibits entropic ordered superfluidity. The system spontaneously breaks global $U(1)$ rotational symmetry, resulting in a non-zero order parameter $\bar{x} > 0$ in the high-temperature limit. To characterize the dynamics, we employ the Martin–Siggia–Rose (MSR) formalism and derive the single-particle Fokker–Planck equation (FPE). By transforming the FPE operator into a Hermitian operator $H$ using the equilibrium Boltzmann distribution $P_{\mathrm{eq}}$, we obtain analytic expressions for the relaxation times $\tau$. Numerical results show that, in the ordered phase ($\bar{x} \neq 0$), the slowest relaxation modes arise from coupled angular and radial fluctuations. These findings demonstrate how interacting bosons with unbounded degrees of freedom can stabilize long-range order even at arbitrarily high temperatures.