Undular Diffusion in Nonlinear Sigma Models

We discuss general features of charge transport in nonrelativistic classical field theories invariant under non-Abelian unitary Lie groups by examining the full structure of two-point dynamical correlation functions in grand-canonical ensembles at finite charge densities (polarized ensembles). Upon explicit breaking of non-Abelian symmetry, two distinct transport laws characterized by dynamical exponent $z=2$ arise. While in the unbroken symmetry sector, the Cartan fields exhibit normal diffusion, the transversal sectors governed by the nonlinear analogs of Goldstone modes disclose an unconventional law of diffusion, characterized by a complex diffusion constant and undulating patterns in the spatiotemporal correlation profiles. In the limit of strong polarization, one retrieves the imaginary-time diffusion for uncoupled linear Goldstone modes, whereas for weak polarizations the imaginary component of the diffusion constant becomes small. In models of higher rank symmetry, we prove absence of dynamical correlations among distinct transversal sectors.

Figure 1

Dynamical correlation functions in the transversal sector, computed in a nonintegrable space-time lattice discretization of $S^2$ Landau-Lifshitz field theory [34], immersed in a longitudinal magnetic field of magnitude $b$ (pointing in the $z$ direction). We display $⟨S^x(x,t)S^x(0,0){⟩}_{\mu }$ evaluated in a grand-canonical ensemble at infinite temperature and chemical potential $\mu =5$ ($⟨S^z⟩\approx 0.8$), shown in absolute value (time step $\tau =1$, length $L=1024$, average over $7.5\times {10}^5$ iterations). Three types of dynamical patterns can be discerned: (a) elliptic regime ($b=-6\times {10}^{-3}$), (b) parabolic regime of undular diffusion without a field ($b=0$), and (c) hyperbolic regime ($b=6\times {10}^{-3}$). The characteristic curves resemble conic sections associated with linear Goldstone modes.

Figure 2

Stationary asymptotic profiles of the transversal Goldstone mode in the nonintegrable space-time lattice discretization of the $S^2$ Landau-Lifshitz field theory [34] without a field ($b=0$), displaying (a) real and (b) imaginary components of ${\lim }_{t\to \infty }|t{|}^{1/2}⟨S^{+}(x,t)S^{-}(0,0){⟩}_{\mu }/2$ as a function of the scaled variable $\xi =t^{-1/2}x$ and U(1) chemical potential $\mu$ [using same parameters as in Fig. 1]. Gray curves mark the prediction of the linear theory, cf. Eq. (2) (in arbitrary units). Dashed blue lines show best two-parameter fits to Eq. (9). (c) Dependence of real and imaginary components of complex diffusion constant $\mathfrak{D}$ on chemical potential $\mu$. Symmetry points $H$ and $G$ designate the U(1)-invariant vacuum and SU(2)-invariant equilibrium measure, respectively (red dashed line is a guide to the eye).