Coupled hydrodynamics in dipole-conserving quantum systems

We investigate the coupled dynamics of charge and energy in interacting lattice models with dipole conservation. We formulate a generic hydrodynamic theory for this combination of fractonic constraints and numerically verify its applicability to the late-time dynamics of a specific bosonic quantum system by developing a microscopic nonequilibrium quantum field theory. Employing a self-consistent $1/N$ approximation in the number of field components, we extract all entries of a generalized diffusion matrix and determine their dependence on microscopic model parameters. We discuss the relation of our results to experiments in ultracold atom quantum simulators.

Figure 1

Coupled hydrodynamics in fractonic quantum matter. (a) Illustration of the dipole-conserving dynamics within our microscopic field theory. Bosons (blue circles) situated on lattice sites need to coordinate with other bosons in order to move. This coordinated hopping, as well as the local boson-boson repulsion, are mediated by exchange field propagators $D$ and $O$, respectively. (b) Decay of an energy excitation including a small charge-density excitation obtained from dipole-conserving hydrodynamics. A diffusive regime with dynamical exponent $z=2$ emerges at intermediate times and crosses over to a subdiffusive regime with $z=4$ at late times. The latter arises from the charge contribution to the excitation and is a signature of dipole conservation. Insets: Spatial profiles of the excitation at intermediate (blue) and late (red) times.

Figure 2

Evolution of a nonequilibrium initial state with a sharply peaked charge excitation. (a) Charge profile at various times. (b) The rescaled charge profile collapses to the scaling function $F^{\left(1\right)}$. (c) The local excitation decays algebraically with exponent close to $1/4$. The short-time dynamics agree with exact results obtained up to quadratic order.

Figure 3

Evolution of nonequilibrium initial states with a homogeneous charge profile and a sharply peaked energy profile. (a) Energy profile at various times. (b) The rescaled energy profiles collapse to a Gaussian $F^{\left(0\right)}$. Here the excitation has been normalized to unity. (c) The excitation decays algebraically with exponent close to 1/2.

Figure 4

Diffusion matrix. Numerical values of the infinite temperature diffusion coefficients as a function of the local interaction strength $U/J$ and filling fraction $n_B$. (a) Charge-charge diffusion constant $D_{nn}$ governing the subdiffusive decay of charge excitations. Inset: $D_{nn}$ as a function of filling for two values of $U/J$. (b) Charge-energy diffusion constant as computed from Onsager relations. (c) Off-diagonal energy-charge diffusion constant. (d) Energy-energy diffusion constant. Dashed lines are obtained by applying Eq. (A5) to linearly interpolated simulation data.

Figure 5

Labeling of lattice sites and bonds in a system of size $L$. Lattice sites and bonds are numbered from left to right beginning with 0. The first lattice site may also be labeled as $1_{-}$ since it is to the left of the first bond, similarly $2_{+}$ refers to the third lattice site.