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Abstract:

The Boltzmann distribution encodes our subjective knowledge of the configuration in a classical lattice model, given only its Hamiltonian. If we acquire further information about the configuration from measurement, our knowledge is updated according to Bayes' theorem. We examine the resulting “conditioned ensembles,” finding that they show many new phase transitions and new renormalization-group fixed points. (Similar conditioned ensembles also describe “partial quenches” in which some of the system's degrees of freedom are instantaneously frozen, while the others continue to evolve.) After describing general features of the replica field theories for these problems, we analyze the effect of measurement on illustrative critical systems, including: critical Ising and Potts models, which show surprisingly rich phase diagrams, with RG fixed points at weak, intermediate, and infinite measurement strength; various models involving free fields, XY spins, or flux lines in 2D or 3D; and geometrical models such as polymers or clusters. We also give a formalism for measurement of classical stochastic processes. We use this to make connections with quantum dynamics, in particular with “charge sharpening” in 1D, for which we give a purely hydrodynamic derivation of the known effective field theory. We discuss qualitative differences between RG flows for the above measured systems, described by $N\to 1$ replica limits, and those for disordered systems, described by $N\to 0$ limits. In addition to discussing measurement of critical states, we give a unifying treatment of a family of inference problems for noncritical states. These are related to the Nishimori line in the phase diagram of the random-bond Ising model, and are relevant to various quantum error correction problems. We describe distinct physical interpretations of conditioned ensembles and note interesting open questions.

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.