From Stochastic Spin Chains to Quantum Kardar-Parisi-Zhang Dynamics

We introduce the asymmetric extension of the quantum symmetric simple exclusion process which is a stochastic model of fermions on a lattice hopping with random amplitudes. In this setting, we analytically show that the time-integrated current of fermions defines a height field that exhibits quantum nonlinear stochastic Kardar-Parisi-Zhang dynamics. Similarly to classical simple exclusion processes, we further introduce the discrete Cole-Hopf (or Gärtner) transform of the height field that satisfies a quantum version of the stochastic heat equation. Finally, we investigate the limit of the height field theory in the continuum under the celebrated Kardar-Parisi-Zhang scaling and the regime of almost-commuting quantum noise.

Figure 1

Representation of the lattice and the hopping process. A fermion can jump to the right (to the left) with probability per unit time $d{W}_t$ ($d{\overline{W}}_t$). These amplitudes correspond to an effective interaction with a bosonic bath.

Figure 2

Mapping from the particle picture to the height field. Bottom. Particles hop on the lattice and the occupied sites are black filled. Top. The configuration of particles is represented by an interface incremented with the presence of a particle at the corresponding site. The particle transport maps to a corner growth process: if a particle hops to the left (to the right), a corner is raised (lowered).