Dr Adam Nahum

Abstract:

We consider the universal aspects of two problems: (i) the singular value structure of a product $M_t=m_t{m}_{t-1}\dots m_1$ of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices $m_i$ , representing time steps in the evolution, but the $m_i$ are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.

Abstract:

Studies of random unitary circuits have shown that the calculation of Rényi entropies of entanglement can be mapped to classical statistical mechanics problems in spacetime. In this paper, we develop an analogous spacetime picture of entanglement generation for random free or weakly interacting fermion systems without conservation laws. We first study a free-fermion model, namely a one-dimensional chain of Majorana modes with nearest-neighbor hoppings, random in both space and time. We analyze the $N\mathrm{th}$ Rényi entropy of entanglement using a replica formalism, and we show that the effective model is equivalent to an $\mathrm{SO}\left(2N\right)$ Heisenberg spin chain evolving in imaginary time. By applying a saddle-point approximation to the coherent states path integral for the $N=2$ case, we arrive at a semiclassical picture for the dynamics of the entanglement purity, in terms of two classical fields in spacetime. The classical solutions involve a smooth domain wall that interpolates between two values, with the width of this smooth domain wall spreading diffusively in time. We then study how adding weak interactions to the free-fermion model modifies this spacetime picture. Interactions reduce the symmetry of the effective continuum description. As a result the width of the entanglement domain wall remains finite, rather than growing diffusively in time. This yields a crossover from diffusive to ballistic spreading of information.